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On topologies on the underlying set of a topological monoid induced by its unitary extensions

On topologies on the underlying set of a topological monoid induced by its unitary extensions Topol. Algebra Appl. 2022; 10:25–35 Research Article Open Access Boris G. Averbukh* On topologies on the underlying set of a topological monoid induced by its unitary extensions https://doi.org/10.1515/taa-2020-0110 Received 9 April, 2021; accepted 7 February, 2022 Abstract: Extensions of a given topological monoid where all its unitary Cauchy lters converge, can induce dierent topologies on its underlying set. We study properties of these topologies and prove a condition under which the initial topology of this monoid is one of them. Keywords: topological monoid, Cauchy lter, completion MSC: 22A15, 54H10 Introduction In analysis, one often uses fundamental sequences of points of a normed linear space, i.e. sequences pos- sessing the property that all dierences of their far enough members lie in any preassigned neighborhood of zero. In paper [1], we dened nets in arbitrary Hausdor topological monoids which have a similar property. We call them unitary Cauchy nets or shortly C-nets and the corresponding lters C-lters and prove that the underlying set of this monoid endowed with the family of all such lters forms a Cauchy space. In papers [2] and [3], using known theorems on completions of Cauchy spaces, we construct extensions of a given monoid X where all its C-nets converge. These extensions are said to be unitary extensions of this monoid. The proposed in [3] constructions are simpler than compactications. However, there is not always such an extension which is itself a topological monoid containing X as a submonoid. Moreover, topologies of unitary extensions can induce dierent topologies on the underlying set of a given monoid. If such an induced topology coincides with the initial one, then the corresponding extension is said to be precise. Paper [3] contains a construction of a precise unitary extension for a monoid satisfying some additional conditions. In paper [4], we use this theory to nd a necessary and sucient condition of the existence of an embedding of a monothetic monoid into a topological group. In the rst section of this paper, we prove a necessary and sucient condition of the existence of unitary extensions with a preassigned induced topology and, in the second one, a sucient condition of the existence of a precise extension which only uses local properties of the given monoid. *Corresponding Author: Boris G. Averbukh: Bauman Moscow State Technical University, department of mathematics, Moscow, Russian Federation, E-mail: averbuch@gmx.de Open Access. © 2022 Boris G. Averbukh, published by De Gruyter. This work is licensed under the Creative Commons Attribution 4.0 License. 26 Ë Boris G. Averbukh 1 A criterion of the existence of unitary extensions with a given induced topology A) In order to make this paper more self-contained, we begin from the reminder of the main concepts and statements of the previous papers [1], [2] and [3] of this series. Let X = (X, m, τ) be a Hausdor topological monoid with an identity 1. Here, m is a multiplication and τ a topology on the underlying set X. In the following, we always shorten m(a, b) to ab. A net S = fx g in X is called a (two-sided) Cauchy (or shortly a C-) net if, for each neighborhood U of α2A 0 0 0 1, there exists α 2 A and, for each α ≥ α , there exists α 2 A such that x 0 2 Ux U for all α ≥ α . Here 0 0 α 0 0 and later, the line on top denotes the topological closure. Throwing away this line, we obtain a denition of a (two-sided) strict C-net. Denitions of left and right C-nets and strict C-nets are similar (see [1]). The lters corresponding to C-nets are said to be C-lters. Their direct denition is: a lter F on X (i.e. in the power set P(X)) is called a C-lter on X if the set M (F) = fx 2 X : UxU 2 Fg belongs to F for any neighborhood U of 1. To obtain a similar denition of a strict C-lter, it is necessary to use sets M (F) = fx 2 X : UxU 2 Fg. If X is a topological group, then right (left, two-sided) C-lters are Cauchy lters of the right (left, Rölke) uniformity on X and any C-lter is a strict one. In the following, we only consider two-sided C-lters and strict C-lters. As Example 1.2 from [3] shows, strict C-lters are not necessarily a more convenient object of study than non-strict ones, and we consider them in parallel. For properties of (strict) C-lters, we refer the reader to papers [1], [2] and [3]. Their summary can be found at the beginning of [3]. In particular, we writeF ≥ F for C-ltersF ,F if M (F ) 2 F for any neighborhood 1 2 1 2 U 2 1 U of 1. The corresponding condition for C-nets S = fx g and S = fy g means: for each neighborhood 1 α2A 2 β β2B U of 1, there exists α 2 A such that, for each α ≥ α , there exists β 2 B such that the inclusion y 2 Ux U 0 0 0 holds for any β ≥ β . We set F ≈ F if both F ≥ F and F ≥ F are true. For strict C-lters F and F , we 0 1 2 1 2 2 1 1 2 s s write F ≥ F if M (F ) 2 F for any neighborhood U of 1. These strict C-lters are said to be s-equivalent 1 2 2 1 s s s (F ≈ F ) if F ≥ F and F ≥ F hold. It is proved in [1] that ≥ is a quasi-order relation, and ≈ is an 1 2 1 2 2 1 equivalence on the set Σ of all C- lters. If F , F are C-lters with F ≥ F and F clusters to a point x , then 1 2 1 2 1 0 F can not have any cluster points diering from x . Moreover, if the topology τ is T , then these conditions 2 0 3 imply that both lters converge to x . For strict C-lters, this statement can be strengthened in the following way: if F , F are strict C-lters such that F ≥ F , and F clusters to x , then F converges to x . 1 2 1 2 1 0 2 0 The intersection of any family of equivalent to each other C-lters is a C-lter from the same equivalence class. If this family is wholly an equivalence class, then its intersection is said to be least with respect to inclusion or shortly-least C-lter. If it is equivalent to a given C-lter F, then we denote it by F . lst For any x 2 X, the symbol x˙ denotes the strict C-lter consisting of all subsets of X containing this point x. Example 1.1. Let X be (R , +) with the topology of Sorgenfrey line (a base at each point x consists of intervals [x; x + ϵ[ with ϵ > 0). Two-least C-lters can be assigned to x ≠ 0, and there are no other-least C-lters. The rst of them is the ultralter x˙ and the second one (we denote it by F ) has a base consisting of intervals [x − ϵ; x[. This C-lter F is strict, does not converge and is ≥ x˙ . It is proved in [1] that the set X endowed with the family Σ is a Cauchy space (for denitions of these spaces and their completions, see for example papers [5] - [12]). It is proved in [1] that the convergence structure of this spase denes a T -topology. This topology is said to be unitary. For any x 2 X, the lter x˙ is the lst neighborhoods lter of x in this topology. Monoid X is said to be unitarily separable if its initial topology τ is coarser than the unitary one or coincides with it. X always possesses this property if its topology τ is T . The underlying set X endowed with the family Σ of strict C-lters is a Cauchy space, too, and its con- vergence structure denes a T -topology. It is ner than the topology which is dened by the family Σ, or coincides with it. Moreover, it is always ner than the initial one or coincides with it, and, in this case, the notion of the unitary separability loses its sense. The Wyler completion X of the Cauchy space (X, Σ) is a Hausdor topological space (see [2]) whose topol- ogy τ˜ is said to be natural. The set X consists of equivalence classes of C-lters or (that is equivalent) of all On topologies on the underlying set of a topological monoid induced by its unitary extensions Ë 27 -least C-lters. Its point corresponding to such a lter F is denoted by [F]. There exists a canonical topo- logical embedding i with i(x) = [x˙ ] of X endowed with the unitary topology into X. The subspace i(X) is lst ˜ ˜ dense and open, and X \ i(X) is discrete. A base of τ˜ at a point [F] 2 X \ i(X) consists of all sets of the form [F][i(V) where V runs open in the unitary topology sets from F. Any lter i(F) (i.e. the lter on X with a base ˜ ˜ i(B) where B is a base of F) with F 2 Σ converges in X to the point [F ], and that is why any point of X is the lst limit of such a lter. Similar statements are true for strict C-lters. Let now Y be a topological space and f : X ! Y an embedding of the underlying set X such that lim f(F) consists of an only point for each C-lter F on X and each point of Y is the limit of such a lter. Then the couple (f , Y) is called a unitary extension of X. Replacing C-lters by strict C-lters, we obtain a denition of a weakly unitary extension. For example, the couple (i, X) satises these requirements. Moreover, it is proved in [2] that, for any ˜ ˜ unitary extension (f , Y) of X, there exists a unique continuous map f : X ! Y such that f = f  i. This map ˜ ˜ is surjective. Thus, (i, X) has a universal property over other unitary extensions. That is why (i, X) is called the nest unitary extension of X. For a given unitary extension (f , Y) of X, denote by τ the topology on X induced by the topology of Y. This topology is said to be extensible by means of f . For example, the unitary topology on X is extensible by means of i if Y is X endowed with the natural topology. It is evident, f is a homeomorphic embedding of the space (X, τ ) into Y. The unitary extension is said to be precise if τ coincides with the initial topology τ of X. It is said to be homomorphic if Y is the underlying space of a Hausdor topological monoid Y and f is a continuous identity preserving homomorphism of X into Y. A precise homomorphic unitary extension is an algebraic and topological embedding. For commutative case, such extensions were studied in [3]. Some monoids (for example, (R , +) with the topology of Sorgenfrey line) do not have them. Therefore, we study arbitrary unitary extensions. The next statement was proved in [3] (see Proposition 2.2). Proposition 1.2. (i) Precise unitary extensions can only exist for unitarily separable monoids. If (f , Y) is an arbitrary unitary extension of X, then: (ii) (X, τ ) is a T -space. f 1 (iii) The topology τ is coarser than the unitary topology or coincides with it. (iv) The limits of any equivalent C-lters in the topology τ coincide. (v) Any C-lter on X either converges in the topology τ to an only point which is its unique cluster point, or does not cluster. Similar to (ii) - (v) statements are true for strict C-lters and weakly unitary extensions. In (iv), equivalent has to be replaced by s-equivalent. B) In this section, our purpose is to nd conditions of the existence of a unitary extension with a given extensible topology. This extension can be obtained from the nest one by means of topological operations, and we begin with the study of an appropriate topology on the set X. Denition 1.3. We say a topology on X is quasi-extensible if it possesses properties (ii), (iii), (iv) and (v) of the topology τ from Proposition 1.2. If the initial topology τ of X is T , then it is quasi-extensible. Indeed, this topology is unitary separable and satises conditions (iv) and (v) by Corollary 1.10 from [1]. Let now τ be a quasi-extensible topology on X. Set −1 τ˜ = fV 2 τ˜ : i (V) is open in the topology τ on Xg. 0 0 It is straightforward that it is a topology on X which is coarser than τ˜. −1 Proposition 1.4. A subset V  X is open in the topology τ˜ i the subset i (V) is open in the topology τ and 0 0 belongs to each-least C-lter F such that [F] 2 V \ i(X). Proof. Necessity. If V  X is open in the natural topology, then, by Lemma 1.10 from [2], for any [F] 2 V \i(X), −1 there exists U 2 F such that i(U)  V. Hence, i (V) 2 F. −1 Suciency. The set i (V) is open in the unitary topology on X since this topology is ner than τ by −1 Denition 1.3. Therefore, i(i (V)) = V \ i(X) is open in the natural topology by Theorem 1.9 from [2]. Let now 28 Ë Boris G. Averbukh −1 [F] 2 V\i(X). By Lemma 1.10 from [2], the set i(i (V))[f[F]g is a neighborhood of [F] in the natural topology. It is evident, this neighborhood is contained in V, and that’s why V is open in the natural topology. Corollary 1.5. (i) The map i : (X, τ ) ! (X, τ˜ ) is a dense homeomorphic embedding. 0 0 (ii) The remainder X\i(X) is closed and discrete in the topology τ˜ . A base of neighborhoods of an arbitrary point [F] 2 X \ i(X) in this topology consists of sets of the formf[F]g[ i(U) where U is open in the topology τ and belongs to F. (iii) (X, τ˜ ) is a T -space. 0 1 −1 Proof. (i) The map i is continuous in these topologies since i (V) is open in the topology τ on X for each V which is open in the topology τ˜ . To prove that this map is open, consider an arbitrary τ -open U  X and 0 0 apply the previous proposition to V = i(U). Moreover, i is injective, and it implies that it is a homeomorphic embedding. The subset i(X) is dense in X in the topology τ˜ since it is dense in the natural topology which is ner than τ˜ . (ii) An arbitrary set A  X \ i(X) is closed in the topology τ˜ since X 2 F for any lter F on X and, ˜ ˜ therefore, the complement V = X\A = i(X)[(X\(i(X)[A)) is open in this topology. Let now V be an arbitrary −1 neighborhood of a point [F] 2 X \ i(X) in the topology τ˜ . Then U = i (V) is open in the topology τ and 0 0 belongs to F. Hence,f[F]g[ i(U)  V is a neighborhood of [F] in the topology τ˜ . (iii) It follows from i) that every singleton f[F]g from X \ i(X) is closed in the topology τ˜ . Show that −1 V = X \fi(x)g is open in this topology for any x 2 X. Indeed, i (V) = X \fxg is open in the topology τ by Denition 1.3. Moreover, this set is a member of any-least C-lter F with [F] 2 X \ i(X) since this lter is not equivalent to the lter x˙ and, therefore, some its member does not contain x. Remark 1.6. The topology τ˜ is not necessarily Hausdor. For example, if x 2 X and [F] 2 X\ i(X) are chosen so that the lter F converges to x in the topology τ , then any neighborhoods of i(x) and [F] in the topology τ˜ have a non-empty intersection. Similarly, if F , F are divergent in the topology τ -least C-lters such 0 1 2 0 that U \ U ≠ ; for any open in this topology U 2 F , U 2 F , then any neighborhoods of [F ] and [F ] 1 2 1 1 2 2 1 2 have a non-empty intersection. For example, it holds for lters F (z ) and F (z ) from Example 2.9. from [1]. 3 0 4 0 Let (f , Y) be a unitary extension of X such that τ = τ . According to the universal property of the nest f 0 unitary extension, there exists a unique continuous in the natural topology τ˜ map f : X ! Y such that f = f  i. This map is surjective and takes each point [F] 2 X to the only point of Y being the limit of the lter f(F) (see Theorem 2.12 from [2]). In the following, we use the next properties of f . Lemma 1.7. (i) The map f is continuous in the topology τ˜ . (ii) For any [F] 2 X and x 2 X, the equality f([F]) = f(x) is true i lim F = x. Here, lim denotes the limit τ τ 0 0 in the topology τ ; by Proposition 1.2 (v), it can consist of at most one point. −1 −1 −1 Proof. (i) If U is an arbitrary open subset of Y, then V = f (U) is open in the topology τ˜ and i (V) = f (U) is open in the topology τ . (ii) The map f : (X, τ ) ! Y is a homeomorphic embedding. Therefore, the equalities f([F]) = lim f(F) = f(x) and lim F = x are equivalent. Now, we need to use the following notions and the following theorem from [3]. Let (f , Y ), (f , Y ) be 1 1 2 2 unitary extensions of X. We write (f , Y ) ≥ (f , Y ) if there exists a continuous map h : Y ! Y such that 1 1 2 2 1 2 h f = f . This map h is unique since each point y 2 Y , i = 1, 2, can be represented in the form y = lim f (F) 1 2 i i where F is a suitable C-lter on X. For such unitary extensions, the extensible topology τ is ner than the extensible topology τ or these topologies coincide. By the universal property of the nest unitary extension, (i, X) ≥ (f , Y) for any unitary extension (f , Y) of X. Unitary extensions (f , Y ), (f , Y ) are said to be equivalent if both the inequalities (f , Y ) ≥ (f , Y ) 1 1 2 2 1 1 2 2 and (f , Y ) ≥ (f , Y ) hold. In this case, the corresponding maps h : Y ! Y and h : Y ! Y are 2 2 1 1 1 1 2 2 2 1 canonical reciprocal homeomorphisms, and such unitary extensions induce the same extensible topology. It is proved in Theorem 2.5 from [3] that any non-empty subset of the ordered set of equivalence classes of unitary extensions has a unique least upper bound. Its extensible topology is the least upper bound of the extensible topologies corresponding to classes from this subset. In particular, for each extensible topology τ , there exists a greatest class of unitary extensions with this extensible topology. ˜ ˜ ˜ ˜ ˜ Denote now by Y the quotient space (X, τ˜ )/E(f) where E(f) is the generated by f equivalence on X, and 0 0 0 let f : (X, τ˜ ) ! Y be the corresponding quotient map and j : Y ! Y the standard continuous bijection. 0 On topologies on the underlying set of a topological monoid induced by its unitary extensions Ë 29 0 0 −1 0 0 The map f = f  i = j  f of (X, τ ) into Y is continuous, and f and f coincide as maps between sets if we identify the sets Y and Y by means of j. 0 0 Proposition 1.8. (i) The map f is a homeomorphism of the space (X, τ ) onto an open dense subspace of Y , 0 0 and the subset Y \ f (X) is discrete. 0 0 0 0 (ii) The couple (f , Y ) is a unitary extension ofX with the same extensible topology τ , and (f , Y ) ≥ (f , Y). Proof. (i) The map f is continuous and injective since f is injective by the above denition of a unitary exten- sion. Therefore, we should only check that it is open. For an arbitrary open in the topology τ subset U  X, consider the set −1 ˜ ˜ V = f (f(U)) = i(U)[f[F] 2 X \ i(X): f([F]) 2 f(U)g. It is saturated by E(f). Show that it is open in the topology τ˜ . In order to use Proposition 1.4, observe that −1 i (V) = U, and [F] 2 V \ i(X) means lim F 2 U by the previous Lemma. It implies U 2 F, and Proposition 0 0 0 0 1.4 is applicable to V. Therefore, f (U) = f (V) is open in Y by the above denition of the topology of Y . The 0 0 subspace f (X) is dense in Y since i(X) is dense in (X, τ˜ ). 0 0 0−1 To prove the second statement of (i), take an arbitrary A  Y \ f (X). Then the subset f (A) of X \ i(X) is closed in the topology τ˜ by Corollary 1.5. Hence, A is closed in the quotient topology of Y . (ii) Let F be an arbitrary C-lter on X. Since j is continuous, the lter f (F) either converges to the point −1 j (lim f(F)) or diverges. Show that the latter case is impossible. By Proposition 1.4 from [2], the lter i(F) converges in the natural topology on X. Since this topology is ner than τ˜ , this lter converges in the topology 0 0 τ˜ although its limit can consist of many points. Therefore, the lter f (i(F)) = f (F) converges, too. Any point 0 0 −1 from Y is the limit of some lter of the form f (F) since j is surjective. By (i), the extensible topology of the 0 0 0 0 unitary extension (f , Y ) is τ again. The inequality (f , Y ) ≥ (f , Y) follows from the continuity of j. The proven proposition shows that, in our search of conditions of the existence of unitary extensions with a given extensible topology, we can only consider the case when the map f is quotient. Denition 1.9. Let (f , Y) be a unitary extension with the extensible topology τ and f : (X, τ˜ ) ! Y the unique 0 0 ˜ ˜ continuous map with f = f  i. This unitary extension is said to be prime if this map f is quotient. Corollary 1.10. The greatest of unitary extensions with a given extensible topology is prime. Now, we can slightly simplify our task. Instead of quotient maps of the space (X, τ˜ ), we shall consider such maps of its subspace ˜ ˜ X (τ ) = i(X)[f[F] 2 X \ i(X): F diverges in the topology τ g. r 0 0 ˜ ˜ Denote by r the map (X, τ˜ ) ! X (τ ) such that r([F]) = i(x) if lim F = x for some x 2 X, and r([F]) = [F] r τ 0 0 0 otherwise. Remind that τ is quasi-extensible, and so the set lim F consists of at most one point. 0 0 ˜ ˜ Proposition 1.11. (i) For any quasi-extensible topology τ , X (τ ) is an open subspace of (X, τ˜ ), and the map 0 0 0 r is a retraction and a quotient map. ˜ ˜ If (f , Y) is a unitary extension of X with τ = τ and f the corresponding map X onto Y, then denote by fj f 0 ˜ ˜ ˜ ˜ the restriction of this map to the subspace X (τ ) and by E(fj) the equivalence on X (τ ) corresponding to fj. r r 0 0 Then: −1 ˜ ˜ (ii)fj (fj(˜x)) = ˜x for each ˜x 2 i(X). ˜ ˜ (iii) f is a quotient map i fj possesses this property. ˜ ˜ ˜ ˜ (iv) The quotient spaces (X, τ˜ )/E(f) and X (τ )/E(fj) are canonically homeomorphic. 0 0 Proof. (i) By Corollary 1.5, each containing i(X) subset of X is open in the topology τ˜ . Show that r(i(x)) = i(x) for any x 2 X. If [F] = i(x), then F = x˙ . This lter converges to x in the unitary topology on X since it is lst a neighborhoods lter of this point in this topology. Hence, it converges to x in the topology τ since this topology is coarser than the unitary one, and r([F]) = i(x). ˜ ˜ Prove now that r is continuous. Let V be an open subset of X (τ ). Then V is open in X in the topology τ˜ . 0 0 −1 −1 It implies i (V) is open in the topology τ . The set r (V) consists of V and of all points [F] 2 X \ i(X) such −1 −1 −1 −1 −1 −1 that lim F 2 i (V). Therefore, i (r (V)) = i (V) 2 F for every [F] 2 r (V) \ i(X). Hence, r (V) is open in the topology τ˜ by Proposition 1.4. To prove that r is a quotient map, denote by E(r) the equivalence on X corresponding to r and take an ˜ ˜ arbitrary open saturated by E(r) subset V of (X, τ˜ ). Then r(V) = V \ X (τ ) since r is a retraction, and r(V) 0 0 is open in X (τ ). 0 30 Ë Boris G. Averbukh (ii) The statement follows immediately from Lemma 1.7 (ii) and the denition of X (τ ). ˜ ˜ ˜ (iii), (iv) By Lemma 1.7 and since r is a retraction, the inclusion E(r)  E(f) is true. Hence, f = fj  r. ˜ ˜ Moreover, by (i), the spaces (X, τ˜ )/E(r) and X (τ ) are canonically homeomorphic. Therefore, our statements 0 0 follow from familiar properties of quotient maps (for example, see [13], Corollaries 2.4.4 and 2.4.5): 1) the composition of quotient maps is a quotient map, 2) if the composition g  f of two continuous maps is a quotient map, then g is a quotient map, too. Lemma 1.12. Let τ be a quasi-extensible topology on X, Y a topological space, y 2 Y, f : (X, τ ) ! Y a 0 0 homeomorphic embedding and g : X (τ ) ! Y a quotient map such that f = gi. (In particular, these conditions are satised if (f , Y) is a prime unitary extension of X with the extensible topology τ and g = fj.) If U runs the −1 family of all open subsets of (X, τ ) such that U 2 F for any [F] 2 g (y), then f(U)[fyg runs a base of Y at the point y. −1 Proof. First, we prove that g (f(U)[fyg) is open in X (τ ) for any open U  (X, τ ) satisfying the condition 0 0 −1 above. Denote V(U) = i(U)[ g (y). It is evident that g(V(U)) = f(U)[fyg. We will show that V(U) is open in −1 ˜ ˜ X (τ ) and coincides with g (f(U)[fyg). Check that V(U) is open in (X, τ˜ ). In order to apply Proposition 1.4, 0 0 −1 −1 −1 observe that i (V(U)) = U. It is evident if i(X)\g (y) = ;. If there exists x 2 X such that i(x) = [x˙ ] 2 g (y), lst −1 then U 2 x˙  x˙ and x 2 U, i.e. i (V(U)) = U again. By the previous proposition, we can conclude that lst V(U) is open in X (τ ). Show now that V(U) is saturated by g. Assume it is false. Then i(U) is not saturated by g, and there exists [F] 2 X (τ )\ i(X) such that g([F]) = f(x) for some x 2 U. The equality [F] = lim i(F) implies [F] 2 lim i(F) 0 τ˜ τ˜ since the topology τ˜ on X is coarser than the natural topology τ˜. Applying g, we obtain f(x) 2 lim f(F) and x 2 lim F since f is a homeomorphic embedding. Hence, x = lim F by the denition of a quasi-extensible τ τ 0 0 ˜ ˜ topology. However, it is impossible for points [F] 2 X (τ ) \ i(X) by the denition of the space X (τ ) above. r r 0 0 Thus, if U satises the conditions from the formulation of the lemma, then g(V(U)) = f(U)[fyg is open in Y since g is a quotient map. −1 Consider now an arbitrary neighborhood W of y and denote U = f (W). We need to prove that U 2 F −1 for any [F] 2 g (y). Indeed, for such F, it follows from [F] 2 lim i(F) that y 2 lim f(F), W 2 f(F) and τ˜ U 2 F. Lemma 1.13. Let (f , Y) be a prime unitary extension of X with the extensible topology τ and f : (X, τ˜ ) ! Y, 0 0 ˜ ˜ ˜ fj: X (τ ) ! Y the corresponding quotient maps. Then f = f  i = fj  i is a homeomorphic embedding of (X, τ ) into Y and the remainder Y \ f(X) is closed and discrete. A base at each point y 2 Y \ f(X) consists of sets of the form f(U)[fyg where U  X is open in the topology τ and belongs to any-minimal C-lter F such −1 −1 ˜ ˜ that [F] 2 f (y) = fj (y). Proof. This statement joins Proposition 1.8 and Lemma 1.12 together. C) Now, everything is ready for the proof of the main theorem of this section. We prove a necessary and sucient condition when there exists a unitary extension of X that induces a given topology on X. Theorem 1.14. If there exists a unitary extension of a Hausdor topological monoid X that induces a given topology τ on its underlying set X, then this topology is quasi-extensible (see Proposition 1.2). Now, let τ be quasi-extensible. (i) If there exists a unitary extension of X with the extensible topology τ , then there exists an equivalence E ˜ ˜ on X (τ ) such that the quotient space Y = X (τ )/E is the space of a unitary extension (f , Y) with the extensible r r 0 0 topology τ . In this case, the corresponding map f is the composition of the topological embedding i : (X, τ ) ! 0 0 ˜ ˜ X (τ ) and the quotient map X (τ ) ! Y. r r 0 0 (ii) Let E be an equivalence on X (τ ). It has the previous property if and only if it satises the following requirements: 1) for any x 2 X, the singletonfi(x)g is a class of E; 2) for any class H of E lying outside i(X) and for any point [F] 2 X not belonging to H (including all points ˜ ˜ from X \ X (τ )), there exists a closed in the topology τ on X set F 2 F such that the set X \ F belongs to each 0 0 -least C-lter H with [H] 2 H. Proof. (i) The statement follows from Corollary 1.10, Proposition 1.11 and the above-mentioned fact that, for any extensible topology, there exists the greatest unitary extension with this extensible topology. On topologies on the underlying set of a topological monoid induced by its unitary extensions Ë 31 (ii) Necessity. As above, let (f , Y) be a prime unitary extension of X with the extensible topology τ and f ˜ ˜ ˜ ˜ and fj the corresponding quotient maps of X and, respectively, of X (τ ) onto Y. Set E = E(fj). It was proved −1 ˜ ˜ in Proposition 1.11 that fj (fj(i(x))) = i(x) for any x 2 X, i.e. 1) holds. −1 Prove 2). Denote H = fj (y) with y 2 Y \ f(X) and H the set of -least C-lters on X corresponding to points from H. Let H be the intersection of these lters and F an arbitrary -least C-lter on X such that ˜ ˜ ˜ ˜ fj([F]) ≠ y. We permit, in particular, that fj([F]) is not dened, i.e. [F] 2 X\X (τ ). In this case, F converges in the topology τ . In any case, f(F) does not cluster to y by the denition of the map f above and by Proposition 1.2. Therefore, there exist a set F 2 f(F) and a neighborhood W of y with the empty intersection. We can 0 00 00 assume that F = f(F ) with F 2 F and W = f(U) [ fyg where the set U 2 H is open in the topology τ (see Lemma 1.12 above). Then F \ U = ;, and it means that F = X \ U 2 F. Suciency. Denote by Y the quotient space X (τ )/E and by g the corresponding quotient map. Set f = gi and prove that (f , Y) is a unitary extension of X with the extensible topology τ . By Corollary 1.5, f is continuous in the topology τ . Condition 1) implies that it is injective. Moreover, it follows from this condition that, for any [F] 2 X (τ ), g([F]) belongs to f(X) only if [F] 2 i(X). Check that f is ˜ ˜ open. Let U  (X, τ ) be open. Then i(U) is open in X (τ ) since it is open in (X, τ˜ ) by the same Corollary. 0 0 0 Moreover, i(U) is saturated by E by condition 1), and, therefore, f(U) = g(i(U)) is open in Y. Thus, f is a homeomorphic embedding of (X, τ ). Show now that f is a unitarily extending map. LetF be an arbitrary C-lter onX andF the corresponding lst -least C-lter. If [F ] 2̸ X (τ ), then F converges in the topology τ , and it follows from F  F that its lst 0 lst 0 lst limit in this topology is contained in the limit of F. This latter limit consists of an only point by Denition 1.3 since the topology τ is quasi-extensible, and we denote y(F) = f(lim F) 2 f(X) \ lim f(F). Let now 0 0 ˜ ˜ [F ] 2 X (τ ). Then [F ] 2 lim i(F) since i(F) converges to [F ] in the natural topology on X and the lst 0 lst τ˜ lst topology τ˜ is coarser than the natural one. In this case, we set y(F) = g([F ]) 2 lim f(F). In particular, 0 lst y(F) = f(x) if [F ] = i(x), and y(F) 2̸ f(X) if [F ] 2 X (τ ) \ i(X). It is evident that, for any point y 2 Y, there lst lst 0 exists F such that y = y(F). Now, it remains to prove that y(F) is the only cluster point of f(F). Let y be another one. Observe that it does not belong to f(X). Otherwise, F (and then also F ) would converge in (X, τ ) by Denition 1.3 since lst 0 τ is a quasi-extensible topology and f is a homeomorphism. It can be only possible if [F ] 2̸ X (τ ) or 0 lst 0 [F ] 2 i(X). In both these cases, f(F) would have two dierent cluster points y(F) and y in f(X), and that’s lst why F would have two such points in X. However, it is impossible by Denition 1.3. −1 0 Denote now H = g (y )  X (τ )\i(X). Then [F ] 2̸ H. By condition ii), there exist a set F 2 F  F and 0 lst lst an open in the topology τ subset U  X such that U 2 H and F\U = ;. By Lemma 1.12, the set f(U)[fy g 0 0 is a neighborhood of y . It does not intersect the set f(F) 2 f(F) since y 2̸ f(X) and f is an embedding. Hence, y cannot be a cluster point of f(F). As the rst application of the previous theorem, we give another proof of the above-mentioned and proven in [3] statement that, for each extensible topology τ , there exists a greatest unitary extensions with this extensible topology. Proposition 1.15. Let τ be a quasi-extensible topology and the set of equivalences on X (τ ) possessing prop- 0 0 erties 1) and 2) above be non-empty. Then: ˜ ˜ (i) The intersection (in X (τ ) × X (τ )) of any family of such equivalences belongs to this set again; r r 0 0 (ii) The intersection of all equivalences from this set denes (see statement (i) of the previous theorem) the greatest unitary extension with the extensible topology τ . Proof. (i) Let fE g be such a family and E = E . It is evident that condition 1) is satised for E. To α α α2A α2A prove 2), take a class H of E lying outside i(X) and a point [F] 2 X not belonging to H. Then, for any α 2 A, there exists class H of E containing H and H = H . For some α, [F] 2̸ H and there exists a closed α α α α α2A in the topology τ set F 2 F such that X \ F belongs to each-least C-lter H with [H] 2 H . It is evident, it implies that X \ F 2 H for any-least C-lter H with [H] 2 H. (ii) Let (f , Y) be a unitary extension of X with the extensible topology τ . By Propositions 1.8 and 1.11, 0 0 0 0 0 ˜ ˜ there exists an equivalence E on X (τ ) such that the couple (f , Y ) where Y is the quotient space X (τ )/E r r 0 0 0 0 0 and f is the composition of the map i and the corresponding quotient map g : X (τ ) ! Y , is a unitary 0 0 0 extension of X with the extensible topology τ and (f , Y ) ≥ (f , Y). By the previous theorem, E possesses 0 32 Ë Boris G. Averbukh ˆ ˜ properties 1) and 2), and it implies that E contains the intersection E of all equivalences on X (τ ) possessing these properties. ˆ ˜ ˆ ˜ ˆ Denote now Y = X (τ )/E, and let g be the quotient maps X (τ ) ! Y. Set f = g  i. By the previous r r 0 0 ˆ ˆ theorem, (f , Y) is a unitary extension of X with the extensible topology τ . The inclusion E  E implies 0 0 0 that there exists a continuous map h : Y ! Y such that g = h  g. Therefore, f = h  f holds, too, and 0 0 (f , Y) ≥ (f , Y ). Proposition 1.16. Let the initial topology τ of X be T and all divergent in this topology -least C-lters have ˜ ˜ bases consisting of closed sets. Then the subspace X (τ) of the space (X, τ˜) is the space Y of the greatest precise unitary extension of X and the corresponding map f coincides with i. Proof. It was observed above that the topology τ is quasi-extensible if it is T . Denote by E the trivial equiv- ˜ ˜ alence on X (τ) whose classes are singletons. It is contained in each equivalence on X. To use the previous Proposition, we only need to check that E satises condition 2) of Theorem 1.14. Keeping the denotations used in its formulation, assume that H consists of an only point [H] where H is a divergent -least C-lter. It does not cluster by Corollary 1.10 from [1] and has a base of closed sets. If F converges to some x, let G 2 H and a neighborhood W of x be chosen such that G \ W = ;. Denote F = W. Then X \ F 2 H. Consider now the case, when F is divergent, too, and so has a base consisting of closed sets. Let condition 2) be not satised for these H and [F] 2̸ H. Then each closed member of F has a non-empty intersection with each member of G. It implies that any members of F and G have non-empty intersections. Therefore, there exists a lter containing both these lters. Hence, F ≈ G by Proposition 2.2 from [1], and F = G because of the uniqueness of-least C-lter in each class of the equivalence relation ≈. We got a contradiction. Example 1.17. Find the greatest precise unitary extension of the monoidX = (Q , +) of non-negative rationals endowed with the usual topology which we denote by τ. For each positive real x, let F be the-least C-lter having a base consisting of sets of the form Q \ [a; x[ where a is rational and a < x. It converges to x if x is rational, and it diverges for irrational x. In the latter case, the base above consists of closed sets. Other- least C-lters are trivial ultralters, so that the previous statement is applicable. The greatest precise unitary + + extension of (Q , +) (i.e. space X (τ) by Proposition 1.16) can be identied with R endowed with a non- 0 0 standard topology which can be found by means of Lemma 1.13. Its trace on Q is τ, the subset of irrationals is closed and discrete, and a base at an arbitrary irrational point x consists of sets (Q \]a; x[)[fxg where a is rational and a < x. Another construction of unitary extensions of monoids whose divergent-least C-lters have bases con- + + sisting of closed sets, was given in [3]. For (Q , +) with the usual topology, it leads to R with the usual 0 0 topology. Thus, the greatest unitary extension with a given extensible topology is not necessarily the best one. Remark 1.18. Analogs of all proven in this section statements are true for strict C-lters and weakly unitary ˜ ˜ extensions. To get them, it is only necessary to replace everywhere the space X with the space X , its subspace ˜ ˜ X (τ) with the subspace of X which has a similar denition, and the equivalence ≈ of C-lters with the s- equivalence ≈ of strict C-lters. In the rest, all our proofs remain unchanged since they only use common properties of all C-lters. 2 A sucient condition of the existence of a precise unitary extension A) The existence of a precise unitary extension of a given topological monoid depends on its local properties. To formulate the proven in this section condition, we need some additional terminology. In the following, we write U  V for subsets U, V from X if there exists a neighborhood O of 1 such that OxO  V for any x 2 U. As above, X denotes the underlying set of the considered Hausdor topological monoid X with the topology τ. On topologies on the underlying set of a topological monoid induced by its unitary extensions Ë 33 Lemma 2.1. Suppose that, for any x 2 X and for any its neighborhood V, there exists a neighborhood U of x such that U  V holds. Then the topology τ is quasi-extensible. Proof. Check the requirements of Denition 1.3. Requirement (ii) ((X, τ) is a T -space) is satised since X is Hausdor. Prove (iii) (τ is coarser than the unitary topology on X or coincides with it). Indeed, let V be an arbitrary neighborhood of this point x, U its neighborhood from the formulation above and O a neighborhood of 1 such that OyO  V for any y 2 U. The neighborhoods lter of the point x in the unitary topology is the lter x˙ which is generated by sets Δ (x) = fy 2 IxI : x 2 IyIg where I runs all neighborhoods of 1 in the lst I topology τ (see Lemma 2.13 from [1]). It is evident that Δ (x)  OxO  V. To prove (iv) (equivalent C-lters have equal limits), let F , F be C-lters, F ≈ F , x = limF , and V , U, O such as above. It follows from 1 2 1 2 1 U 2 F and M (F ) 2 F that U\ M (F ) ≠ ;, and there exists a point y 2 U such that OyO 2 F . Therefore, 1 O 2 1 O 2 2 V 2 F and limF = x, too. To prove (v) (any C-lter either converges in the topology τ or does not cluster), 2 2 let V , U, O be such as above, and C-lter F clusters to x. Then U \ M (F) ≠ ;, and there exists a point y 2 U such that OyO 2 F. Therefore, V 2 F, limF = x, and F has not any other cluster points, since (X, τ) is Hausdor. Denition 2.2. Let U, V be open subsets of (X, τ) with U  V. The couple (U, V) is said to be innitely subdivisible if there exists a familyfW g (n, k 2 N are non-negative integers, k = 0, 2 ) of open subsets of n,k 0 (X, τ) satisfying the following conditions: a) W = U, W = V. 0 0 0 1 n−1 b) W = W for any n 2 N, k = 0, 2 . In particular, W = U, W n = V for any n 2 N. n 2k n−1 k n 0 n 2 c) W  W for any n 2 N , k = 1, 2 . n k−1 n k 0 We use below that, for any innitely subdivisible couple (U, V), the corresponding couples (U, W ), (W , W ) and (W , V) are innitely subdivisible, too. 21 22 22 Denition 2.3. The monoid X is said to be everywhere innitely subdivisible if, for any x 2 X and for any neighborhood V of x, there exists its neighborhood U such that U  V and the couple (U, V) is innitely subdivisible. It was suciently to assume here that this condition is only satised for any x 2 X and for any V from some base at the point x. The requirement to be everywhere innitely subdivisible is stronger than the continuity of the multipli- cation in the considered monoid. For example, the topological monoid from Example 2.9 from [1] does not possess this property. However, it is weaker than its uniform continuity. Indeed, let P and Q be open en- tourages of its uniformity such that Q  P + R for some its entourage R, and S be an open entourage such that 2S  R. Denote by U, V , W the neighborhoods of an arbitrary x 2 X corresponding, respectively, to 0 0 P, Q, P + S. There exists a neighborhood O of 1 such that (x , y) 2 S for any x 2 X and for any y 2 Ox O. 0 0 0 0 Therefore, Ox O  W for any x 2 U, and Ox O  V for any x 2 W, i.e. U  W  V. It is evident that this process can be limitlessly continued. On the other hand, the monoid X = (Q , ·) with the additive uniformity is everywhere innitely subdivisible but non-uniform. Theorem 2.4. If a given Hausdor topological monoid is everywhere innitely subdivisible, then there exists its precise unitary extension. Proof. We keep the notation used in the proof of Theorem 1.14. The topology τ is quasi-extensible by Lemma 2.1, and we only need to prove that there exists an equivalence on the space X (τ) satisfying conditions 1) and 2) of this Theorem. Instead of equivalences, we will consider partitions of subspaces of this space satisfying some conditions. We will dene an order relation on the set of such partitions and will prove that this set has a maximal element and the equivalence corresponding to this element possesses the necessary properties. We will call a subspace Y of X (τ) admissible if it contains i(X). A partition P = fP g of an admissible r α α2A subspace Y is said to be admissible if it satises the following conditions: a) Each singletonfi(x)g with x 2 X is an element of P; b) For any element P of P, for any-least C-lter F such that [F] 2 P , and for any innitely subdivisible α α couple (U, V) of open subsets of X, if U 2 F then V 2 P where P is the set of -least C-lters on X α α corresponding to points from P . The latter condition concerns elements of P consisting of points corresponding to divergent (in the topol- ogy τ) -least C-lters. For any such element, these lters must form an equi-divergent at each point x 2 X 34 Ë Boris G. Averbukh family: if (U, V) is a innitely subdivisible couple of neighborhoods of x and V does not belong to one of these lters, then U does not belong to any of them. The set H of couples (Y , P) where Y is an admissible subspace of X (τ) and P is its admissible partition, is non-empty since it contains i(X) together with its partition in singletons. It follows from the fact that U  V for any innitely subdivisible couple (U, V). Endow H with an order relation. Let Y , Y be admissible subspaces and P , P their admissible parti- 1 2 1 2 tions. Set (Y , P ) ≤ (Y , P ) if each class of P is contained in some class of P . It implies that Y  Y . 1 1 2 2 1 2 1 2 We prove now that (H, ≤) has a maximal element. Let L be a linearly ordered subset of H. For each h = (Y , P) 2 H, denote Y = Y , P = P . Set Y = Y and dene its partition P in the following h h h h2L way. For each [F] 2 Y, let P ([F]) denote its class in the partition P . If [F] 2 / Y , then P ([F]) = ;. Set h h h h now P([F]) = P ([F]) and show that the totality of all the sets of the form P([F]) is a partition of Y. If h2L P([F ])\ P([F ]) ≠ ; for some [F ], [F ] 2 Y, then there exist h , h 2 L such that P ([F ])\ P ([F ]) ≠ ;. 1 2 1 2 1 2 h 1 h 2 1 2 Assume that h ≤ h . Then P ([F ])\ P ([F ]) ≠ ; for any h ≥ h . It implies P ([F ]) = P ([F ]) for h ≥ h and 1 2 h 1 h 2 2 h 1 h 2 2 P([F ]) = P ([F ]) = P([F ]). 1 h 1 2 h2L, h≥h It is evident that Y is admissible and P possesses the property a). Check the property b). Let P = P([F]), (U, V) be an innitely subdivisible couple, U 2 F and [G] 2 P([F]) for some -least C-lter G. Then there exists h 2 L such that [G] 2 P ([F]). Therefore, V 2 G since P is admissible, and P possesses the property h h b), too. Thus, (Y , P) 2 H and it is evident that (Y , P ) ≤ (Y , P) for any h 2 L. Now, the existence of a maximal h h element of (H, ≤) follows from Zorn’s lemma. ˜ ˜ Observe that Y = X (τ) for any maximal element (Y , P) 2 H. Indeed, if there exists a point ˜x 2 X (τ)\ Y, r r then the subspace Y = Y [f˜xg is admissible and its partition consisting of classes of P and of singletonf˜xg 0 0 0 0 is admissible, too, i.e. (Y , P ) 2 H and (Y , P) ≤ (Y , P ). Now, we move on to the last part of the proof: we will establish that the equivalence on X (τ) which corresponds to an arbitrary maximal element (Y , P) 2 H, possesses the properties 1) and 2) of Theorem 1.14. The properties 1) and a) coincide, and we only need to check 2). Suppose it is false, i.e. there exist a class ˜ ˜ P 2 P lying in X (τ)\ i(X) and a-least C-lter F with [F] 2 X \ P such that F\ T ≠ ; for any closed (in the α r α topology τ) set F 2 F and for any set T 2 P . As above, P is the set of -least C-lters corresponding to α α points from P . First, we show that this lter F diverges. Suppose it converges to x , and let G be an arbitrary -least C-lter such that [G] 2 P . The lter G diverges, and, therefore, there exists a neighborhood V of x such that V 2 / G. Denote by U a neighborhood of x such that the couple (U, V) is innitely subdivisible. For any 0 0 0 -least C-lter G with [G ] 2 P , U does not belong to G by condition b) above. It implies that X\U intersects 0 0 any member of G and belongs to some lter containing G . By Proposition 2.2 from [1], this is a C-lter which is equivalent to G , and it follows from Lemma 2.12 from [1] that the set (X\ U) := fx 2 X : OxO\ (X\ U) ≠ ;g 0 0 0 belongs to G for any neighborhood O of 1 and for any G with [G ] 2 P . Choose a neighborhood O of 1 and a neighborhood W of x so that OxO  U for any x 2 W. It is possible since X is everywhere innitely subdivisible. Denote by O a neighborhood of 1 such that O  O. The set W belongs to F, and it implies that W \ M (F) ≠ ;. Set F = O xO where x belongs to this intersection. Then F \ (X \ U) = ; although F is O 1 1 O 1 1 closed and belongs to F and T = (X \ U) belongs to P . The fact that F diverges in the topology τ, implies that [F] 2 X (τ). Therefore, there exists its class P([F]) of the partition P. Denote now by P the partition which arises from P if we unite its classes P and P([F]). 0 0 ˜ ˜ ˜ Then (X (τ), P) ≤ (X (τ), P ), and we will obtain a contradiction if we will prove that (X (τ), P ) 2 H. r r r 0 0 It is evident that P satises the condition a). To check b), we only need to consider the class P ([F]). Let (U, V) be an innitely subdivisible couple of open sets and G a -least C-lter such that [G] 2 P ([F]) and U 2 G. For these U and V, choose open sets W , W as in Denition 2.2 and denote by O a neighborhood 2 1 2 2 of 1 such that O x O  W for any x 2 W . 2 2 2 1 We need to consider two cases: either [G] 2 P([F]) or [G] 2 P . First, we assume [G] 2 P([F]). Then U 2 G 0 0 implies W 2 F and W \M 0(F) ≠ ; for any neighborhood O of 1. Therefore, O xO 2 F for some x 2 W . 2 1 2 1 O 2 1 For suciently small O , we can assume that there exists a neighborhood I of 1 such that IO I  O. For this S S T 0 0 0 0 x, O xO \ M (G ) ≠ ; since M (G ) 2 P and any closed set from F intersects any set from 0 0 α I I [G ]2P [G ]2P α α On topologies on the underlying set of a topological monoid induced by its unitary extensions Ë 35 0 0 0 0 0 0 0 0 P . Hence, O xO \ M (G ) ≠ ; for some G with [G ] 2 P . For these x and G , there exists y 2 O xO such α α T T 0 0 0 that IyI 2 G , and it implies that OxO 2 G . Therefore, W 2 G and V 2 P . Moreover, V 2 P([F]), and 2 2 that is why V 2 P ([F]). Let now [G] 2 P . Then U 2 G implies W 2 P . Therefore, M (F)\ W ≠ ; since W is open and α α 2 1 O 2 1 2 1 T T M (F) 2 F, and O x O 2 F for some x 2 W . Hence, W 2 F, V 2 P([F]) and V 2 P ([F]). O 2 1 2 2 Remark 2.5. Let (f , Y) be a precise unitary extension of X. Then lim f(F) consists of an only point for any strict C-lter F. Therefore, the subspace of Y consisting of all limits of such a form is a precise weakly unitary extension of this monoid. Hence, the condition of Theorem 2.4 is also sucient for the existence of such an extension. The author does not know how to weaken this condition. Now, we return to the concept of an everywhere innitely subdivisible monoid. Let V be an arbitrary open subset of (X, τ). For any neighborhood O of 1, denote by T (O) the set fx 2 X : OxO  Vg. It is evident, T (O )  T (O ) for any O , O with O  O . Consider the next condition V 1 V 2 1 2 2 1 (A): for any open V and for any neighborhood O of 1, there exists its neighborhood O such that T (O ) 1 2 V 1 Int T (O ). Here, Int M denotes the interior of the subspace M of (X, τ). V 2 Proposition 2.6. If this condition A is satised, then any couple (U, V) of open subsets with U  V is innitely subdivisible. Proof. Indeed, U  V means that there is a neighborhood O of 1 such that U  T (O). Let O be a neighbor- V 1 hood of 1 such that O  O. For any suitable O , set W = Int T (O ). Then O O xO O  OxO  V for any 2 V 2 1 1 1 1 x 2 U. It implies that O xO  T (O )  W for any x 2 U and U  W. Moreover, W  T (O ) and W  V 1 1 V 1 V 2 take place, too. For example, if the identity 1 has a neighborhood with a compact closure, then the above condition is satised since T (O) is open for all suciently small O. Indeed, if O is compact, then it is evident that OxO = OxO. For any x 2 T (O), o , o 2 O, there exist neighborhoods U = U (o , o , x) of o , U = U (o , o , x) V 1 2 1 1 1 2 1 2 2 1 2 of o and W = W(o , o , x) of x such that U WU  V. The family of open sets of the form U × U is a cover 2 1 2 1 2 1 2 of the compact set O× O  X× X, and we can select a nite subcover. Then the intersection of sets of the form W(o , o , x) corresponding to elements of this subcover is a neighborhood of x lying in T (O). 1 2 V If (X, τ) is a T -space, then it is evident that, for any neighborhood V of any point x 2 X, there exists its neighborhood U such that U  V holds. Therefore, if X additionally satises the condition A, then it is everywhere innitely subdivisible. References [1] B. G. Averbukh, On unitary Cauchy lters on topological monoids, Topol. Algebra Appl., 1 (2013), 46-59. [2] B. G. Averbukh, On nest unitary extensions of topological monoids, Topol. Algebra Appl.,3 (2015), 1-10. [3] B. G. Averbukh, On unitary extensions and unitary completions of topological monoids, Topol. Algebra Appl., 4 (2016), 18-30. [4] B. G. Averbukh, A criterion of the existence of an embedding of a monothetic monoid into a topological group, Topol. Algebra Appl., 7(2019), 1-12. [5] R. Fric, D. C. Kent, Completion functors for Cauchy spaces, Int. J. Math. & Math. Sci. 2, No. 4 (1979), 589-604. MR 80#54042. Zbl 428.54018. [6] H. H. Keller, Die Limes-Uniformisierbarkeit der Limesräume, Math. Ann. 176 (1968), 334-341. MR 37#874. Zbl 155.50302 [7] D. C. Kent, G.D. Richardson, Regular completions of Cauchy spaces, Pacic Journal of mathematics 51 No. 2 (1974), 483-490. [8] D. C. Kent, G. D. Richardson, Cauchy completion categories, Canad. Math. Bull. 32, No. 1 (1989), 78-84. MR 90#54007. Zbl 675.54004. [9] N. Rath, Completion of a Cauchy space without the T -restriction on the space, Internat. J. Math. & Math. Sci. 24, No. 3 (2000), 163-172. [10] N. Rath, Completions of Filter Semigroups, Acta Math. Hungar. 107 (1-2) (2005), 45-54. [11] E. E. Reed, Completions of Uniform Convergence Spaces, Math. Ann. 194 (1971), 83-108. MR 45#1109. Zbl 217.19603. [12] O. Wyler, Ein Komplettierungsfunktor für uniforme Limesräume, Math. Nachr. 46 (1970), 1-12. MR 44#985. Zbl 207.52603. [13] R. Engelking, General topology. Rev. and compl. ed., Sigma Series in Pure Mathematics, 6., Berlin: Heldermann Verlag, 1989, viii + 529 pp., ISBN 3-88538-006-4, Zbl 0684.54001. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Topological Algebra and its Applications de Gruyter

On topologies on the underlying set of a topological monoid induced by its unitary extensions

Averbukh, Boris G.
Topological Algebra and its Applications , Volume 10 (1): 11 – Jan 1, 2022
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Abstract

Topol. Algebra Appl. 2022; 10:25–35 Research Article Open Access Boris G. Averbukh* On topologies on the underlying set of a topological monoid induced by its unitary extensions https://doi.org/10.1515/taa-2020-0110 Received 9 April, 2021; accepted 7 February, 2022 Abstract: Extensions of a given topological monoid where all its unitary Cauchy lters converge, can induce dierent topologies on its underlying set. We study properties of these topologies and prove a condition under which the initial topology of this monoid is one of them. Keywords: topological monoid, Cauchy lter, completion MSC: 22A15, 54H10 Introduction In analysis, one often uses fundamental sequences of points of a normed linear space, i.e. sequences pos- sessing the property that all dierences of their far enough members lie in any preassigned neighborhood of zero. In paper [1], we dened nets in arbitrary Hausdor topological monoids which have a similar property. We call them unitary Cauchy nets or shortly C-nets and the corresponding lters C-lters and prove that the underlying set of this monoid endowed with the family of all such lters forms a Cauchy space. In papers [2] and [3], using known theorems on completions of Cauchy spaces, we construct extensions of a given monoid X where all its C-nets converge. These extensions are said to be unitary extensions of this monoid. The proposed in [3] constructions are simpler than compactications. However, there is not always such an extension which is itself a topological monoid containing X as a submonoid. Moreover, topologies of unitary extensions can induce dierent topologies on the underlying set of a given monoid. If such an induced topology coincides with the initial one, then the corresponding extension is said to be precise. Paper [3] contains a construction of a precise unitary extension for a monoid satisfying some additional conditions. In paper [4], we use this theory to nd a necessary and sucient condition of the existence of an embedding of a monothetic monoid into a topological group. In the rst section of this paper, we prove a necessary and sucient condition of the existence of unitary extensions with a preassigned induced topology and, in the second one, a sucient condition of the existence of a precise extension which only uses local properties of the given monoid. *Corresponding Author: Boris G. Averbukh: Bauman Moscow State Technical University, department of mathematics, Moscow, Russian Federation, E-mail: averbuch@gmx.de Open Access. © 2022 Boris G. Averbukh, published by De Gruyter. This work is licensed under the Creative Commons Attribution 4.0 License. 26 Ë Boris G. Averbukh 1 A criterion of the existence of unitary extensions with a given induced topology A) In order to make this paper more self-contained, we begin from the reminder of the main concepts and statements of the previous papers [1], [2] and [3] of this series. Let X = (X, m, τ) be a Hausdor topological monoid with an identity 1. Here, m is a multiplication and τ a topology on the underlying set X. In the following, we always shorten m(a, b) to ab. A net S = fx g in X is called a (two-sided) Cauchy (or shortly a C-) net if, for each neighborhood U of α2A 0 0 0 1, there exists α 2 A and, for each α ≥ α , there exists α 2 A such that x 0 2 Ux U for all α ≥ α . Here 0 0 α 0 0 and later, the line on top denotes the topological closure. Throwing away this line, we obtain a denition of a (two-sided) strict C-net. Denitions of left and right C-nets and strict C-nets are similar (see [1]). The lters corresponding to C-nets are said to be C-lters. Their direct denition is: a lter F on X (i.e. in the power set P(X)) is called a C-lter on X if the set M (F) = fx 2 X : UxU 2 Fg belongs to F for any neighborhood U of 1. To obtain a similar denition of a strict C-lter, it is necessary to use sets M (F) = fx 2 X : UxU 2 Fg. If X is a topological group, then right (left, two-sided) C-lters are Cauchy lters of the right (left, Rölke) uniformity on X and any C-lter is a strict one. In the following, we only consider two-sided C-lters and strict C-lters. As Example 1.2 from [3] shows, strict C-lters are not necessarily a more convenient object of study than non-strict ones, and we consider them in parallel. For properties of (strict) C-lters, we refer the reader to papers [1], [2] and [3]. Their summary can be found at the beginning of [3]. In particular, we writeF ≥ F for C-ltersF ,F if M (F ) 2 F for any neighborhood 1 2 1 2 U 2 1 U of 1. The corresponding condition for C-nets S = fx g and S = fy g means: for each neighborhood 1 α2A 2 β β2B U of 1, there exists α 2 A such that, for each α ≥ α , there exists β 2 B such that the inclusion y 2 Ux U 0 0 0 holds for any β ≥ β . We set F ≈ F if both F ≥ F and F ≥ F are true. For strict C-lters F and F , we 0 1 2 1 2 2 1 1 2 s s write F ≥ F if M (F ) 2 F for any neighborhood U of 1. These strict C-lters are said to be s-equivalent 1 2 2 1 s s s (F ≈ F ) if F ≥ F and F ≥ F hold. It is proved in [1] that ≥ is a quasi-order relation, and ≈ is an 1 2 1 2 2 1 equivalence on the set Σ of all C- lters. If F , F are C-lters with F ≥ F and F clusters to a point x , then 1 2 1 2 1 0 F can not have any cluster points diering from x . Moreover, if the topology τ is T , then these conditions 2 0 3 imply that both lters converge to x . For strict C-lters, this statement can be strengthened in the following way: if F , F are strict C-lters such that F ≥ F , and F clusters to x , then F converges to x . 1 2 1 2 1 0 2 0 The intersection of any family of equivalent to each other C-lters is a C-lter from the same equivalence class. If this family is wholly an equivalence class, then its intersection is said to be least with respect to inclusion or shortly-least C-lter. If it is equivalent to a given C-lter F, then we denote it by F . lst For any x 2 X, the symbol x˙ denotes the strict C-lter consisting of all subsets of X containing this point x. Example 1.1. Let X be (R , +) with the topology of Sorgenfrey line (a base at each point x consists of intervals [x; x + ϵ[ with ϵ > 0). Two-least C-lters can be assigned to x ≠ 0, and there are no other-least C-lters. The rst of them is the ultralter x˙ and the second one (we denote it by F ) has a base consisting of intervals [x − ϵ; x[. This C-lter F is strict, does not converge and is ≥ x˙ . It is proved in [1] that the set X endowed with the family Σ is a Cauchy space (for denitions of these spaces and their completions, see for example papers [5] - [12]). It is proved in [1] that the convergence structure of this spase denes a T -topology. This topology is said to be unitary. For any x 2 X, the lter x˙ is the lst neighborhoods lter of x in this topology. Monoid X is said to be unitarily separable if its initial topology τ is coarser than the unitary one or coincides with it. X always possesses this property if its topology τ is T . The underlying set X endowed with the family Σ of strict C-lters is a Cauchy space, too, and its con- vergence structure denes a T -topology. It is ner than the topology which is dened by the family Σ, or coincides with it. Moreover, it is always ner than the initial one or coincides with it, and, in this case, the notion of the unitary separability loses its sense. The Wyler completion X of the Cauchy space (X, Σ) is a Hausdor topological space (see [2]) whose topol- ogy τ˜ is said to be natural. The set X consists of equivalence classes of C-lters or (that is equivalent) of all On topologies on the underlying set of a topological monoid induced by its unitary extensions Ë 27 -least C-lters. Its point corresponding to such a lter F is denoted by [F]. There exists a canonical topo- logical embedding i with i(x) = [x˙ ] of X endowed with the unitary topology into X. The subspace i(X) is lst ˜ ˜ dense and open, and X \ i(X) is discrete. A base of τ˜ at a point [F] 2 X \ i(X) consists of all sets of the form [F][i(V) where V runs open in the unitary topology sets from F. Any lter i(F) (i.e. the lter on X with a base ˜ ˜ i(B) where B is a base of F) with F 2 Σ converges in X to the point [F ], and that is why any point of X is the lst limit of such a lter. Similar statements are true for strict C-lters. Let now Y be a topological space and f : X ! Y an embedding of the underlying set X such that lim f(F) consists of an only point for each C-lter F on X and each point of Y is the limit of such a lter. Then the couple (f , Y) is called a unitary extension of X. Replacing C-lters by strict C-lters, we obtain a denition of a weakly unitary extension. For example, the couple (i, X) satises these requirements. Moreover, it is proved in [2] that, for any ˜ ˜ unitary extension (f , Y) of X, there exists a unique continuous map f : X ! Y such that f = f  i. This map ˜ ˜ is surjective. Thus, (i, X) has a universal property over other unitary extensions. That is why (i, X) is called the nest unitary extension of X. For a given unitary extension (f , Y) of X, denote by τ the topology on X induced by the topology of Y. This topology is said to be extensible by means of f . For example, the unitary topology on X is extensible by means of i if Y is X endowed with the natural topology. It is evident, f is a homeomorphic embedding of the space (X, τ ) into Y. The unitary extension is said to be precise if τ coincides with the initial topology τ of X. It is said to be homomorphic if Y is the underlying space of a Hausdor topological monoid Y and f is a continuous identity preserving homomorphism of X into Y. A precise homomorphic unitary extension is an algebraic and topological embedding. For commutative case, such extensions were studied in [3]. Some monoids (for example, (R , +) with the topology of Sorgenfrey line) do not have them. Therefore, we study arbitrary unitary extensions. The next statement was proved in [3] (see Proposition 2.2). Proposition 1.2. (i) Precise unitary extensions can only exist for unitarily separable monoids. If (f , Y) is an arbitrary unitary extension of X, then: (ii) (X, τ ) is a T -space. f 1 (iii) The topology τ is coarser than the unitary topology or coincides with it. (iv) The limits of any equivalent C-lters in the topology τ coincide. (v) Any C-lter on X either converges in the topology τ to an only point which is its unique cluster point, or does not cluster. Similar to (ii) - (v) statements are true for strict C-lters and weakly unitary extensions. In (iv), equivalent has to be replaced by s-equivalent. B) In this section, our purpose is to nd conditions of the existence of a unitary extension with a given extensible topology. This extension can be obtained from the nest one by means of topological operations, and we begin with the study of an appropriate topology on the set X. Denition 1.3. We say a topology on X is quasi-extensible if it possesses properties (ii), (iii), (iv) and (v) of the topology τ from Proposition 1.2. If the initial topology τ of X is T , then it is quasi-extensible. Indeed, this topology is unitary separable and satises conditions (iv) and (v) by Corollary 1.10 from [1]. Let now τ be a quasi-extensible topology on X. Set −1 τ˜ = fV 2 τ˜ : i (V) is open in the topology τ on Xg. 0 0 It is straightforward that it is a topology on X which is coarser than τ˜. −1 Proposition 1.4. A subset V  X is open in the topology τ˜ i the subset i (V) is open in the topology τ and 0 0 belongs to each-least C-lter F such that [F] 2 V \ i(X). Proof. Necessity. If V  X is open in the natural topology, then, by Lemma 1.10 from [2], for any [F] 2 V \i(X), −1 there exists U 2 F such that i(U)  V. Hence, i (V) 2 F. −1 Suciency. The set i (V) is open in the unitary topology on X since this topology is ner than τ by −1 Denition 1.3. Therefore, i(i (V)) = V \ i(X) is open in the natural topology by Theorem 1.9 from [2]. Let now 28 Ë Boris G. Averbukh −1 [F] 2 V\i(X). By Lemma 1.10 from [2], the set i(i (V))[f[F]g is a neighborhood of [F] in the natural topology. It is evident, this neighborhood is contained in V, and that’s why V is open in the natural topology. Corollary 1.5. (i) The map i : (X, τ ) ! (X, τ˜ ) is a dense homeomorphic embedding. 0 0 (ii) The remainder X\i(X) is closed and discrete in the topology τ˜ . A base of neighborhoods of an arbitrary point [F] 2 X \ i(X) in this topology consists of sets of the formf[F]g[ i(U) where U is open in the topology τ and belongs to F. (iii) (X, τ˜ ) is a T -space. 0 1 −1 Proof. (i) The map i is continuous in these topologies since i (V) is open in the topology τ on X for each V which is open in the topology τ˜ . To prove that this map is open, consider an arbitrary τ -open U  X and 0 0 apply the previous proposition to V = i(U). Moreover, i is injective, and it implies that it is a homeomorphic embedding. The subset i(X) is dense in X in the topology τ˜ since it is dense in the natural topology which is ner than τ˜ . (ii) An arbitrary set A  X \ i(X) is closed in the topology τ˜ since X 2 F for any lter F on X and, ˜ ˜ therefore, the complement V = X\A = i(X)[(X\(i(X)[A)) is open in this topology. Let now V be an arbitrary −1 neighborhood of a point [F] 2 X \ i(X) in the topology τ˜ . Then U = i (V) is open in the topology τ and 0 0 belongs to F. Hence,f[F]g[ i(U)  V is a neighborhood of [F] in the topology τ˜ . (iii) It follows from i) that every singleton f[F]g from X \ i(X) is closed in the topology τ˜ . Show that −1 V = X \fi(x)g is open in this topology for any x 2 X. Indeed, i (V) = X \fxg is open in the topology τ by Denition 1.3. Moreover, this set is a member of any-least C-lter F with [F] 2 X \ i(X) since this lter is not equivalent to the lter x˙ and, therefore, some its member does not contain x. Remark 1.6. The topology τ˜ is not necessarily Hausdor. For example, if x 2 X and [F] 2 X\ i(X) are chosen so that the lter F converges to x in the topology τ , then any neighborhoods of i(x) and [F] in the topology τ˜ have a non-empty intersection. Similarly, if F , F are divergent in the topology τ -least C-lters such 0 1 2 0 that U \ U ≠ ; for any open in this topology U 2 F , U 2 F , then any neighborhoods of [F ] and [F ] 1 2 1 1 2 2 1 2 have a non-empty intersection. For example, it holds for lters F (z ) and F (z ) from Example 2.9. from [1]. 3 0 4 0 Let (f , Y) be a unitary extension of X such that τ = τ . According to the universal property of the nest f 0 unitary extension, there exists a unique continuous in the natural topology τ˜ map f : X ! Y such that f = f  i. This map is surjective and takes each point [F] 2 X to the only point of Y being the limit of the lter f(F) (see Theorem 2.12 from [2]). In the following, we use the next properties of f . Lemma 1.7. (i) The map f is continuous in the topology τ˜ . (ii) For any [F] 2 X and x 2 X, the equality f([F]) = f(x) is true i lim F = x. Here, lim denotes the limit τ τ 0 0 in the topology τ ; by Proposition 1.2 (v), it can consist of at most one point. −1 −1 −1 Proof. (i) If U is an arbitrary open subset of Y, then V = f (U) is open in the topology τ˜ and i (V) = f (U) is open in the topology τ . (ii) The map f : (X, τ ) ! Y is a homeomorphic embedding. Therefore, the equalities f([F]) = lim f(F) = f(x) and lim F = x are equivalent. Now, we need to use the following notions and the following theorem from [3]. Let (f , Y ), (f , Y ) be 1 1 2 2 unitary extensions of X. We write (f , Y ) ≥ (f , Y ) if there exists a continuous map h : Y ! Y such that 1 1 2 2 1 2 h f = f . This map h is unique since each point y 2 Y , i = 1, 2, can be represented in the form y = lim f (F) 1 2 i i where F is a suitable C-lter on X. For such unitary extensions, the extensible topology τ is ner than the extensible topology τ or these topologies coincide. By the universal property of the nest unitary extension, (i, X) ≥ (f , Y) for any unitary extension (f , Y) of X. Unitary extensions (f , Y ), (f , Y ) are said to be equivalent if both the inequalities (f , Y ) ≥ (f , Y ) 1 1 2 2 1 1 2 2 and (f , Y ) ≥ (f , Y ) hold. In this case, the corresponding maps h : Y ! Y and h : Y ! Y are 2 2 1 1 1 1 2 2 2 1 canonical reciprocal homeomorphisms, and such unitary extensions induce the same extensible topology. It is proved in Theorem 2.5 from [3] that any non-empty subset of the ordered set of equivalence classes of unitary extensions has a unique least upper bound. Its extensible topology is the least upper bound of the extensible topologies corresponding to classes from this subset. In particular, for each extensible topology τ , there exists a greatest class of unitary extensions with this extensible topology. ˜ ˜ ˜ ˜ ˜ Denote now by Y the quotient space (X, τ˜ )/E(f) where E(f) is the generated by f equivalence on X, and 0 0 0 let f : (X, τ˜ ) ! Y be the corresponding quotient map and j : Y ! Y the standard continuous bijection. 0 On topologies on the underlying set of a topological monoid induced by its unitary extensions Ë 29 0 0 −1 0 0 The map f = f  i = j  f of (X, τ ) into Y is continuous, and f and f coincide as maps between sets if we identify the sets Y and Y by means of j. 0 0 Proposition 1.8. (i) The map f is a homeomorphism of the space (X, τ ) onto an open dense subspace of Y , 0 0 and the subset Y \ f (X) is discrete. 0 0 0 0 (ii) The couple (f , Y ) is a unitary extension ofX with the same extensible topology τ , and (f , Y ) ≥ (f , Y). Proof. (i) The map f is continuous and injective since f is injective by the above denition of a unitary exten- sion. Therefore, we should only check that it is open. For an arbitrary open in the topology τ subset U  X, consider the set −1 ˜ ˜ V = f (f(U)) = i(U)[f[F] 2 X \ i(X): f([F]) 2 f(U)g. It is saturated by E(f). Show that it is open in the topology τ˜ . In order to use Proposition 1.4, observe that −1 i (V) = U, and [F] 2 V \ i(X) means lim F 2 U by the previous Lemma. It implies U 2 F, and Proposition 0 0 0 0 1.4 is applicable to V. Therefore, f (U) = f (V) is open in Y by the above denition of the topology of Y . The 0 0 subspace f (X) is dense in Y since i(X) is dense in (X, τ˜ ). 0 0 0−1 To prove the second statement of (i), take an arbitrary A  Y \ f (X). Then the subset f (A) of X \ i(X) is closed in the topology τ˜ by Corollary 1.5. Hence, A is closed in the quotient topology of Y . (ii) Let F be an arbitrary C-lter on X. Since j is continuous, the lter f (F) either converges to the point −1 j (lim f(F)) or diverges. Show that the latter case is impossible. By Proposition 1.4 from [2], the lter i(F) converges in the natural topology on X. Since this topology is ner than τ˜ , this lter converges in the topology 0 0 τ˜ although its limit can consist of many points. Therefore, the lter f (i(F)) = f (F) converges, too. Any point 0 0 −1 from Y is the limit of some lter of the form f (F) since j is surjective. By (i), the extensible topology of the 0 0 0 0 unitary extension (f , Y ) is τ again. The inequality (f , Y ) ≥ (f , Y) follows from the continuity of j. The proven proposition shows that, in our search of conditions of the existence of unitary extensions with a given extensible topology, we can only consider the case when the map f is quotient. Denition 1.9. Let (f , Y) be a unitary extension with the extensible topology τ and f : (X, τ˜ ) ! Y the unique 0 0 ˜ ˜ continuous map with f = f  i. This unitary extension is said to be prime if this map f is quotient. Corollary 1.10. The greatest of unitary extensions with a given extensible topology is prime. Now, we can slightly simplify our task. Instead of quotient maps of the space (X, τ˜ ), we shall consider such maps of its subspace ˜ ˜ X (τ ) = i(X)[f[F] 2 X \ i(X): F diverges in the topology τ g. r 0 0 ˜ ˜ Denote by r the map (X, τ˜ ) ! X (τ ) such that r([F]) = i(x) if lim F = x for some x 2 X, and r([F]) = [F] r τ 0 0 0 otherwise. Remind that τ is quasi-extensible, and so the set lim F consists of at most one point. 0 0 ˜ ˜ Proposition 1.11. (i) For any quasi-extensible topology τ , X (τ ) is an open subspace of (X, τ˜ ), and the map 0 0 0 r is a retraction and a quotient map. ˜ ˜ If (f , Y) is a unitary extension of X with τ = τ and f the corresponding map X onto Y, then denote by fj f 0 ˜ ˜ ˜ ˜ the restriction of this map to the subspace X (τ ) and by E(fj) the equivalence on X (τ ) corresponding to fj. r r 0 0 Then: −1 ˜ ˜ (ii)fj (fj(˜x)) = ˜x for each ˜x 2 i(X). ˜ ˜ (iii) f is a quotient map i fj possesses this property. ˜ ˜ ˜ ˜ (iv) The quotient spaces (X, τ˜ )/E(f) and X (τ )/E(fj) are canonically homeomorphic. 0 0 Proof. (i) By Corollary 1.5, each containing i(X) subset of X is open in the topology τ˜ . Show that r(i(x)) = i(x) for any x 2 X. If [F] = i(x), then F = x˙ . This lter converges to x in the unitary topology on X since it is lst a neighborhoods lter of this point in this topology. Hence, it converges to x in the topology τ since this topology is coarser than the unitary one, and r([F]) = i(x). ˜ ˜ Prove now that r is continuous. Let V be an open subset of X (τ ). Then V is open in X in the topology τ˜ . 0 0 −1 −1 It implies i (V) is open in the topology τ . The set r (V) consists of V and of all points [F] 2 X \ i(X) such −1 −1 −1 −1 −1 −1 that lim F 2 i (V). Therefore, i (r (V)) = i (V) 2 F for every [F] 2 r (V) \ i(X). Hence, r (V) is open in the topology τ˜ by Proposition 1.4. To prove that r is a quotient map, denote by E(r) the equivalence on X corresponding to r and take an ˜ ˜ arbitrary open saturated by E(r) subset V of (X, τ˜ ). Then r(V) = V \ X (τ ) since r is a retraction, and r(V) 0 0 is open in X (τ ). 0 30 Ë Boris G. Averbukh (ii) The statement follows immediately from Lemma 1.7 (ii) and the denition of X (τ ). ˜ ˜ ˜ (iii), (iv) By Lemma 1.7 and since r is a retraction, the inclusion E(r)  E(f) is true. Hence, f = fj  r. ˜ ˜ Moreover, by (i), the spaces (X, τ˜ )/E(r) and X (τ ) are canonically homeomorphic. Therefore, our statements 0 0 follow from familiar properties of quotient maps (for example, see [13], Corollaries 2.4.4 and 2.4.5): 1) the composition of quotient maps is a quotient map, 2) if the composition g  f of two continuous maps is a quotient map, then g is a quotient map, too. Lemma 1.12. Let τ be a quasi-extensible topology on X, Y a topological space, y 2 Y, f : (X, τ ) ! Y a 0 0 homeomorphic embedding and g : X (τ ) ! Y a quotient map such that f = gi. (In particular, these conditions are satised if (f , Y) is a prime unitary extension of X with the extensible topology τ and g = fj.) If U runs the −1 family of all open subsets of (X, τ ) such that U 2 F for any [F] 2 g (y), then f(U)[fyg runs a base of Y at the point y. −1 Proof. First, we prove that g (f(U)[fyg) is open in X (τ ) for any open U  (X, τ ) satisfying the condition 0 0 −1 above. Denote V(U) = i(U)[ g (y). It is evident that g(V(U)) = f(U)[fyg. We will show that V(U) is open in −1 ˜ ˜ X (τ ) and coincides with g (f(U)[fyg). Check that V(U) is open in (X, τ˜ ). In order to apply Proposition 1.4, 0 0 −1 −1 −1 observe that i (V(U)) = U. It is evident if i(X)\g (y) = ;. If there exists x 2 X such that i(x) = [x˙ ] 2 g (y), lst −1 then U 2 x˙  x˙ and x 2 U, i.e. i (V(U)) = U again. By the previous proposition, we can conclude that lst V(U) is open in X (τ ). Show now that V(U) is saturated by g. Assume it is false. Then i(U) is not saturated by g, and there exists [F] 2 X (τ )\ i(X) such that g([F]) = f(x) for some x 2 U. The equality [F] = lim i(F) implies [F] 2 lim i(F) 0 τ˜ τ˜ since the topology τ˜ on X is coarser than the natural topology τ˜. Applying g, we obtain f(x) 2 lim f(F) and x 2 lim F since f is a homeomorphic embedding. Hence, x = lim F by the denition of a quasi-extensible τ τ 0 0 ˜ ˜ topology. However, it is impossible for points [F] 2 X (τ ) \ i(X) by the denition of the space X (τ ) above. r r 0 0 Thus, if U satises the conditions from the formulation of the lemma, then g(V(U)) = f(U)[fyg is open in Y since g is a quotient map. −1 Consider now an arbitrary neighborhood W of y and denote U = f (W). We need to prove that U 2 F −1 for any [F] 2 g (y). Indeed, for such F, it follows from [F] 2 lim i(F) that y 2 lim f(F), W 2 f(F) and τ˜ U 2 F. Lemma 1.13. Let (f , Y) be a prime unitary extension of X with the extensible topology τ and f : (X, τ˜ ) ! Y, 0 0 ˜ ˜ ˜ fj: X (τ ) ! Y the corresponding quotient maps. Then f = f  i = fj  i is a homeomorphic embedding of (X, τ ) into Y and the remainder Y \ f(X) is closed and discrete. A base at each point y 2 Y \ f(X) consists of sets of the form f(U)[fyg where U  X is open in the topology τ and belongs to any-minimal C-lter F such −1 −1 ˜ ˜ that [F] 2 f (y) = fj (y). Proof. This statement joins Proposition 1.8 and Lemma 1.12 together. C) Now, everything is ready for the proof of the main theorem of this section. We prove a necessary and sucient condition when there exists a unitary extension of X that induces a given topology on X. Theorem 1.14. If there exists a unitary extension of a Hausdor topological monoid X that induces a given topology τ on its underlying set X, then this topology is quasi-extensible (see Proposition 1.2). Now, let τ be quasi-extensible. (i) If there exists a unitary extension of X with the extensible topology τ , then there exists an equivalence E ˜ ˜ on X (τ ) such that the quotient space Y = X (τ )/E is the space of a unitary extension (f , Y) with the extensible r r 0 0 topology τ . In this case, the corresponding map f is the composition of the topological embedding i : (X, τ ) ! 0 0 ˜ ˜ X (τ ) and the quotient map X (τ ) ! Y. r r 0 0 (ii) Let E be an equivalence on X (τ ). It has the previous property if and only if it satises the following requirements: 1) for any x 2 X, the singletonfi(x)g is a class of E; 2) for any class H of E lying outside i(X) and for any point [F] 2 X not belonging to H (including all points ˜ ˜ from X \ X (τ )), there exists a closed in the topology τ on X set F 2 F such that the set X \ F belongs to each 0 0 -least C-lter H with [H] 2 H. Proof. (i) The statement follows from Corollary 1.10, Proposition 1.11 and the above-mentioned fact that, for any extensible topology, there exists the greatest unitary extension with this extensible topology. On topologies on the underlying set of a topological monoid induced by its unitary extensions Ë 31 (ii) Necessity. As above, let (f , Y) be a prime unitary extension of X with the extensible topology τ and f ˜ ˜ ˜ ˜ and fj the corresponding quotient maps of X and, respectively, of X (τ ) onto Y. Set E = E(fj). It was proved −1 ˜ ˜ in Proposition 1.11 that fj (fj(i(x))) = i(x) for any x 2 X, i.e. 1) holds. −1 Prove 2). Denote H = fj (y) with y 2 Y \ f(X) and H the set of -least C-lters on X corresponding to points from H. Let H be the intersection of these lters and F an arbitrary -least C-lter on X such that ˜ ˜ ˜ ˜ fj([F]) ≠ y. We permit, in particular, that fj([F]) is not dened, i.e. [F] 2 X\X (τ ). In this case, F converges in the topology τ . In any case, f(F) does not cluster to y by the denition of the map f above and by Proposition 1.2. Therefore, there exist a set F 2 f(F) and a neighborhood W of y with the empty intersection. We can 0 00 00 assume that F = f(F ) with F 2 F and W = f(U) [ fyg where the set U 2 H is open in the topology τ (see Lemma 1.12 above). Then F \ U = ;, and it means that F = X \ U 2 F. Suciency. Denote by Y the quotient space X (τ )/E and by g the corresponding quotient map. Set f = gi and prove that (f , Y) is a unitary extension of X with the extensible topology τ . By Corollary 1.5, f is continuous in the topology τ . Condition 1) implies that it is injective. Moreover, it follows from this condition that, for any [F] 2 X (τ ), g([F]) belongs to f(X) only if [F] 2 i(X). Check that f is ˜ ˜ open. Let U  (X, τ ) be open. Then i(U) is open in X (τ ) since it is open in (X, τ˜ ) by the same Corollary. 0 0 0 Moreover, i(U) is saturated by E by condition 1), and, therefore, f(U) = g(i(U)) is open in Y. Thus, f is a homeomorphic embedding of (X, τ ). Show now that f is a unitarily extending map. LetF be an arbitrary C-lter onX andF the corresponding lst -least C-lter. If [F ] 2̸ X (τ ), then F converges in the topology τ , and it follows from F  F that its lst 0 lst 0 lst limit in this topology is contained in the limit of F. This latter limit consists of an only point by Denition 1.3 since the topology τ is quasi-extensible, and we denote y(F) = f(lim F) 2 f(X) \ lim f(F). Let now 0 0 ˜ ˜ [F ] 2 X (τ ). Then [F ] 2 lim i(F) since i(F) converges to [F ] in the natural topology on X and the lst 0 lst τ˜ lst topology τ˜ is coarser than the natural one. In this case, we set y(F) = g([F ]) 2 lim f(F). In particular, 0 lst y(F) = f(x) if [F ] = i(x), and y(F) 2̸ f(X) if [F ] 2 X (τ ) \ i(X). It is evident that, for any point y 2 Y, there lst lst 0 exists F such that y = y(F). Now, it remains to prove that y(F) is the only cluster point of f(F). Let y be another one. Observe that it does not belong to f(X). Otherwise, F (and then also F ) would converge in (X, τ ) by Denition 1.3 since lst 0 τ is a quasi-extensible topology and f is a homeomorphism. It can be only possible if [F ] 2̸ X (τ ) or 0 lst 0 [F ] 2 i(X). In both these cases, f(F) would have two dierent cluster points y(F) and y in f(X), and that’s lst why F would have two such points in X. However, it is impossible by Denition 1.3. −1 0 Denote now H = g (y )  X (τ )\i(X). Then [F ] 2̸ H. By condition ii), there exist a set F 2 F  F and 0 lst lst an open in the topology τ subset U  X such that U 2 H and F\U = ;. By Lemma 1.12, the set f(U)[fy g 0 0 is a neighborhood of y . It does not intersect the set f(F) 2 f(F) since y 2̸ f(X) and f is an embedding. Hence, y cannot be a cluster point of f(F). As the rst application of the previous theorem, we give another proof of the above-mentioned and proven in [3] statement that, for each extensible topology τ , there exists a greatest unitary extensions with this extensible topology. Proposition 1.15. Let τ be a quasi-extensible topology and the set of equivalences on X (τ ) possessing prop- 0 0 erties 1) and 2) above be non-empty. Then: ˜ ˜ (i) The intersection (in X (τ ) × X (τ )) of any family of such equivalences belongs to this set again; r r 0 0 (ii) The intersection of all equivalences from this set denes (see statement (i) of the previous theorem) the greatest unitary extension with the extensible topology τ . Proof. (i) Let fE g be such a family and E = E . It is evident that condition 1) is satised for E. To α α α2A α2A prove 2), take a class H of E lying outside i(X) and a point [F] 2 X not belonging to H. Then, for any α 2 A, there exists class H of E containing H and H = H . For some α, [F] 2̸ H and there exists a closed α α α α α2A in the topology τ set F 2 F such that X \ F belongs to each-least C-lter H with [H] 2 H . It is evident, it implies that X \ F 2 H for any-least C-lter H with [H] 2 H. (ii) Let (f , Y) be a unitary extension of X with the extensible topology τ . By Propositions 1.8 and 1.11, 0 0 0 0 0 ˜ ˜ there exists an equivalence E on X (τ ) such that the couple (f , Y ) where Y is the quotient space X (τ )/E r r 0 0 0 0 0 and f is the composition of the map i and the corresponding quotient map g : X (τ ) ! Y , is a unitary 0 0 0 extension of X with the extensible topology τ and (f , Y ) ≥ (f , Y). By the previous theorem, E possesses 0 32 Ë Boris G. Averbukh ˆ ˜ properties 1) and 2), and it implies that E contains the intersection E of all equivalences on X (τ ) possessing these properties. ˆ ˜ ˆ ˜ ˆ Denote now Y = X (τ )/E, and let g be the quotient maps X (τ ) ! Y. Set f = g  i. By the previous r r 0 0 ˆ ˆ theorem, (f , Y) is a unitary extension of X with the extensible topology τ . The inclusion E  E implies 0 0 0 that there exists a continuous map h : Y ! Y such that g = h  g. Therefore, f = h  f holds, too, and 0 0 (f , Y) ≥ (f , Y ). Proposition 1.16. Let the initial topology τ of X be T and all divergent in this topology -least C-lters have ˜ ˜ bases consisting of closed sets. Then the subspace X (τ) of the space (X, τ˜) is the space Y of the greatest precise unitary extension of X and the corresponding map f coincides with i. Proof. It was observed above that the topology τ is quasi-extensible if it is T . Denote by E the trivial equiv- ˜ ˜ alence on X (τ) whose classes are singletons. It is contained in each equivalence on X. To use the previous Proposition, we only need to check that E satises condition 2) of Theorem 1.14. Keeping the denotations used in its formulation, assume that H consists of an only point [H] where H is a divergent -least C-lter. It does not cluster by Corollary 1.10 from [1] and has a base of closed sets. If F converges to some x, let G 2 H and a neighborhood W of x be chosen such that G \ W = ;. Denote F = W. Then X \ F 2 H. Consider now the case, when F is divergent, too, and so has a base consisting of closed sets. Let condition 2) be not satised for these H and [F] 2̸ H. Then each closed member of F has a non-empty intersection with each member of G. It implies that any members of F and G have non-empty intersections. Therefore, there exists a lter containing both these lters. Hence, F ≈ G by Proposition 2.2 from [1], and F = G because of the uniqueness of-least C-lter in each class of the equivalence relation ≈. We got a contradiction. Example 1.17. Find the greatest precise unitary extension of the monoidX = (Q , +) of non-negative rationals endowed with the usual topology which we denote by τ. For each positive real x, let F be the-least C-lter having a base consisting of sets of the form Q \ [a; x[ where a is rational and a < x. It converges to x if x is rational, and it diverges for irrational x. In the latter case, the base above consists of closed sets. Other- least C-lters are trivial ultralters, so that the previous statement is applicable. The greatest precise unitary + + extension of (Q , +) (i.e. space X (τ) by Proposition 1.16) can be identied with R endowed with a non- 0 0 standard topology which can be found by means of Lemma 1.13. Its trace on Q is τ, the subset of irrationals is closed and discrete, and a base at an arbitrary irrational point x consists of sets (Q \]a; x[)[fxg where a is rational and a < x. Another construction of unitary extensions of monoids whose divergent-least C-lters have bases con- + + sisting of closed sets, was given in [3]. For (Q , +) with the usual topology, it leads to R with the usual 0 0 topology. Thus, the greatest unitary extension with a given extensible topology is not necessarily the best one. Remark 1.18. Analogs of all proven in this section statements are true for strict C-lters and weakly unitary ˜ ˜ extensions. To get them, it is only necessary to replace everywhere the space X with the space X , its subspace ˜ ˜ X (τ) with the subspace of X which has a similar denition, and the equivalence ≈ of C-lters with the s- equivalence ≈ of strict C-lters. In the rest, all our proofs remain unchanged since they only use common properties of all C-lters. 2 A sucient condition of the existence of a precise unitary extension A) The existence of a precise unitary extension of a given topological monoid depends on its local properties. To formulate the proven in this section condition, we need some additional terminology. In the following, we write U  V for subsets U, V from X if there exists a neighborhood O of 1 such that OxO  V for any x 2 U. As above, X denotes the underlying set of the considered Hausdor topological monoid X with the topology τ. On topologies on the underlying set of a topological monoid induced by its unitary extensions Ë 33 Lemma 2.1. Suppose that, for any x 2 X and for any its neighborhood V, there exists a neighborhood U of x such that U  V holds. Then the topology τ is quasi-extensible. Proof. Check the requirements of Denition 1.3. Requirement (ii) ((X, τ) is a T -space) is satised since X is Hausdor. Prove (iii) (τ is coarser than the unitary topology on X or coincides with it). Indeed, let V be an arbitrary neighborhood of this point x, U its neighborhood from the formulation above and O a neighborhood of 1 such that OyO  V for any y 2 U. The neighborhoods lter of the point x in the unitary topology is the lter x˙ which is generated by sets Δ (x) = fy 2 IxI : x 2 IyIg where I runs all neighborhoods of 1 in the lst I topology τ (see Lemma 2.13 from [1]). It is evident that Δ (x)  OxO  V. To prove (iv) (equivalent C-lters have equal limits), let F , F be C-lters, F ≈ F , x = limF , and V , U, O such as above. It follows from 1 2 1 2 1 U 2 F and M (F ) 2 F that U\ M (F ) ≠ ;, and there exists a point y 2 U such that OyO 2 F . Therefore, 1 O 2 1 O 2 2 V 2 F and limF = x, too. To prove (v) (any C-lter either converges in the topology τ or does not cluster), 2 2 let V , U, O be such as above, and C-lter F clusters to x. Then U \ M (F) ≠ ;, and there exists a point y 2 U such that OyO 2 F. Therefore, V 2 F, limF = x, and F has not any other cluster points, since (X, τ) is Hausdor. Denition 2.2. Let U, V be open subsets of (X, τ) with U  V. The couple (U, V) is said to be innitely subdivisible if there exists a familyfW g (n, k 2 N are non-negative integers, k = 0, 2 ) of open subsets of n,k 0 (X, τ) satisfying the following conditions: a) W = U, W = V. 0 0 0 1 n−1 b) W = W for any n 2 N, k = 0, 2 . In particular, W = U, W n = V for any n 2 N. n 2k n−1 k n 0 n 2 c) W  W for any n 2 N , k = 1, 2 . n k−1 n k 0 We use below that, for any innitely subdivisible couple (U, V), the corresponding couples (U, W ), (W , W ) and (W , V) are innitely subdivisible, too. 21 22 22 Denition 2.3. The monoid X is said to be everywhere innitely subdivisible if, for any x 2 X and for any neighborhood V of x, there exists its neighborhood U such that U  V and the couple (U, V) is innitely subdivisible. It was suciently to assume here that this condition is only satised for any x 2 X and for any V from some base at the point x. The requirement to be everywhere innitely subdivisible is stronger than the continuity of the multipli- cation in the considered monoid. For example, the topological monoid from Example 2.9 from [1] does not possess this property. However, it is weaker than its uniform continuity. Indeed, let P and Q be open en- tourages of its uniformity such that Q  P + R for some its entourage R, and S be an open entourage such that 2S  R. Denote by U, V , W the neighborhoods of an arbitrary x 2 X corresponding, respectively, to 0 0 P, Q, P + S. There exists a neighborhood O of 1 such that (x , y) 2 S for any x 2 X and for any y 2 Ox O. 0 0 0 0 Therefore, Ox O  W for any x 2 U, and Ox O  V for any x 2 W, i.e. U  W  V. It is evident that this process can be limitlessly continued. On the other hand, the monoid X = (Q , ·) with the additive uniformity is everywhere innitely subdivisible but non-uniform. Theorem 2.4. If a given Hausdor topological monoid is everywhere innitely subdivisible, then there exists its precise unitary extension. Proof. We keep the notation used in the proof of Theorem 1.14. The topology τ is quasi-extensible by Lemma 2.1, and we only need to prove that there exists an equivalence on the space X (τ) satisfying conditions 1) and 2) of this Theorem. Instead of equivalences, we will consider partitions of subspaces of this space satisfying some conditions. We will dene an order relation on the set of such partitions and will prove that this set has a maximal element and the equivalence corresponding to this element possesses the necessary properties. We will call a subspace Y of X (τ) admissible if it contains i(X). A partition P = fP g of an admissible r α α2A subspace Y is said to be admissible if it satises the following conditions: a) Each singletonfi(x)g with x 2 X is an element of P; b) For any element P of P, for any-least C-lter F such that [F] 2 P , and for any innitely subdivisible α α couple (U, V) of open subsets of X, if U 2 F then V 2 P where P is the set of -least C-lters on X α α corresponding to points from P . The latter condition concerns elements of P consisting of points corresponding to divergent (in the topol- ogy τ) -least C-lters. For any such element, these lters must form an equi-divergent at each point x 2 X 34 Ë Boris G. Averbukh family: if (U, V) is a innitely subdivisible couple of neighborhoods of x and V does not belong to one of these lters, then U does not belong to any of them. The set H of couples (Y , P) where Y is an admissible subspace of X (τ) and P is its admissible partition, is non-empty since it contains i(X) together with its partition in singletons. It follows from the fact that U  V for any innitely subdivisible couple (U, V). Endow H with an order relation. Let Y , Y be admissible subspaces and P , P their admissible parti- 1 2 1 2 tions. Set (Y , P ) ≤ (Y , P ) if each class of P is contained in some class of P . It implies that Y  Y . 1 1 2 2 1 2 1 2 We prove now that (H, ≤) has a maximal element. Let L be a linearly ordered subset of H. For each h = (Y , P) 2 H, denote Y = Y , P = P . Set Y = Y and dene its partition P in the following h h h h2L way. For each [F] 2 Y, let P ([F]) denote its class in the partition P . If [F] 2 / Y , then P ([F]) = ;. Set h h h h now P([F]) = P ([F]) and show that the totality of all the sets of the form P([F]) is a partition of Y. If h2L P([F ])\ P([F ]) ≠ ; for some [F ], [F ] 2 Y, then there exist h , h 2 L such that P ([F ])\ P ([F ]) ≠ ;. 1 2 1 2 1 2 h 1 h 2 1 2 Assume that h ≤ h . Then P ([F ])\ P ([F ]) ≠ ; for any h ≥ h . It implies P ([F ]) = P ([F ]) for h ≥ h and 1 2 h 1 h 2 2 h 1 h 2 2 P([F ]) = P ([F ]) = P([F ]). 1 h 1 2 h2L, h≥h It is evident that Y is admissible and P possesses the property a). Check the property b). Let P = P([F]), (U, V) be an innitely subdivisible couple, U 2 F and [G] 2 P([F]) for some -least C-lter G. Then there exists h 2 L such that [G] 2 P ([F]). Therefore, V 2 G since P is admissible, and P possesses the property h h b), too. Thus, (Y , P) 2 H and it is evident that (Y , P ) ≤ (Y , P) for any h 2 L. Now, the existence of a maximal h h element of (H, ≤) follows from Zorn’s lemma. ˜ ˜ Observe that Y = X (τ) for any maximal element (Y , P) 2 H. Indeed, if there exists a point ˜x 2 X (τ)\ Y, r r then the subspace Y = Y [f˜xg is admissible and its partition consisting of classes of P and of singletonf˜xg 0 0 0 0 is admissible, too, i.e. (Y , P ) 2 H and (Y , P) ≤ (Y , P ). Now, we move on to the last part of the proof: we will establish that the equivalence on X (τ) which corresponds to an arbitrary maximal element (Y , P) 2 H, possesses the properties 1) and 2) of Theorem 1.14. The properties 1) and a) coincide, and we only need to check 2). Suppose it is false, i.e. there exist a class ˜ ˜ P 2 P lying in X (τ)\ i(X) and a-least C-lter F with [F] 2 X \ P such that F\ T ≠ ; for any closed (in the α r α topology τ) set F 2 F and for any set T 2 P . As above, P is the set of -least C-lters corresponding to α α points from P . First, we show that this lter F diverges. Suppose it converges to x , and let G be an arbitrary -least C-lter such that [G] 2 P . The lter G diverges, and, therefore, there exists a neighborhood V of x such that V 2 / G. Denote by U a neighborhood of x such that the couple (U, V) is innitely subdivisible. For any 0 0 0 -least C-lter G with [G ] 2 P , U does not belong to G by condition b) above. It implies that X\U intersects 0 0 any member of G and belongs to some lter containing G . By Proposition 2.2 from [1], this is a C-lter which is equivalent to G , and it follows from Lemma 2.12 from [1] that the set (X\ U) := fx 2 X : OxO\ (X\ U) ≠ ;g 0 0 0 belongs to G for any neighborhood O of 1 and for any G with [G ] 2 P . Choose a neighborhood O of 1 and a neighborhood W of x so that OxO  U for any x 2 W. It is possible since X is everywhere innitely subdivisible. Denote by O a neighborhood of 1 such that O  O. The set W belongs to F, and it implies that W \ M (F) ≠ ;. Set F = O xO where x belongs to this intersection. Then F \ (X \ U) = ; although F is O 1 1 O 1 1 closed and belongs to F and T = (X \ U) belongs to P . The fact that F diverges in the topology τ, implies that [F] 2 X (τ). Therefore, there exists its class P([F]) of the partition P. Denote now by P the partition which arises from P if we unite its classes P and P([F]). 0 0 ˜ ˜ ˜ Then (X (τ), P) ≤ (X (τ), P ), and we will obtain a contradiction if we will prove that (X (τ), P ) 2 H. r r r 0 0 It is evident that P satises the condition a). To check b), we only need to consider the class P ([F]). Let (U, V) be an innitely subdivisible couple of open sets and G a -least C-lter such that [G] 2 P ([F]) and U 2 G. For these U and V, choose open sets W , W as in Denition 2.2 and denote by O a neighborhood 2 1 2 2 of 1 such that O x O  W for any x 2 W . 2 2 2 1 We need to consider two cases: either [G] 2 P([F]) or [G] 2 P . First, we assume [G] 2 P([F]). Then U 2 G 0 0 implies W 2 F and W \M 0(F) ≠ ; for any neighborhood O of 1. Therefore, O xO 2 F for some x 2 W . 2 1 2 1 O 2 1 For suciently small O , we can assume that there exists a neighborhood I of 1 such that IO I  O. For this S S T 0 0 0 0 x, O xO \ M (G ) ≠ ; since M (G ) 2 P and any closed set from F intersects any set from 0 0 α I I [G ]2P [G ]2P α α On topologies on the underlying set of a topological monoid induced by its unitary extensions Ë 35 0 0 0 0 0 0 0 0 P . Hence, O xO \ M (G ) ≠ ; for some G with [G ] 2 P . For these x and G , there exists y 2 O xO such α α T T 0 0 0 that IyI 2 G , and it implies that OxO 2 G . Therefore, W 2 G and V 2 P . Moreover, V 2 P([F]), and 2 2 that is why V 2 P ([F]). Let now [G] 2 P . Then U 2 G implies W 2 P . Therefore, M (F)\ W ≠ ; since W is open and α α 2 1 O 2 1 2 1 T T M (F) 2 F, and O x O 2 F for some x 2 W . Hence, W 2 F, V 2 P([F]) and V 2 P ([F]). O 2 1 2 2 Remark 2.5. Let (f , Y) be a precise unitary extension of X. Then lim f(F) consists of an only point for any strict C-lter F. Therefore, the subspace of Y consisting of all limits of such a form is a precise weakly unitary extension of this monoid. Hence, the condition of Theorem 2.4 is also sucient for the existence of such an extension. The author does not know how to weaken this condition. Now, we return to the concept of an everywhere innitely subdivisible monoid. Let V be an arbitrary open subset of (X, τ). For any neighborhood O of 1, denote by T (O) the set fx 2 X : OxO  Vg. It is evident, T (O )  T (O ) for any O , O with O  O . Consider the next condition V 1 V 2 1 2 2 1 (A): for any open V and for any neighborhood O of 1, there exists its neighborhood O such that T (O ) 1 2 V 1 Int T (O ). Here, Int M denotes the interior of the subspace M of (X, τ). V 2 Proposition 2.6. If this condition A is satised, then any couple (U, V) of open subsets with U  V is innitely subdivisible. Proof. Indeed, U  V means that there is a neighborhood O of 1 such that U  T (O). Let O be a neighbor- V 1 hood of 1 such that O  O. For any suitable O , set W = Int T (O ). Then O O xO O  OxO  V for any 2 V 2 1 1 1 1 x 2 U. It implies that O xO  T (O )  W for any x 2 U and U  W. Moreover, W  T (O ) and W  V 1 1 V 1 V 2 take place, too. For example, if the identity 1 has a neighborhood with a compact closure, then the above condition is satised since T (O) is open for all suciently small O. Indeed, if O is compact, then it is evident that OxO = OxO. For any x 2 T (O), o , o 2 O, there exist neighborhoods U = U (o , o , x) of o , U = U (o , o , x) V 1 2 1 1 1 2 1 2 2 1 2 of o and W = W(o , o , x) of x such that U WU  V. The family of open sets of the form U × U is a cover 2 1 2 1 2 1 2 of the compact set O× O  X× X, and we can select a nite subcover. Then the intersection of sets of the form W(o , o , x) corresponding to elements of this subcover is a neighborhood of x lying in T (O). 1 2 V If (X, τ) is a T -space, then it is evident that, for any neighborhood V of any point x 2 X, there exists its neighborhood U such that U  V holds. Therefore, if X additionally satises the condition A, then it is everywhere innitely subdivisible. References [1] B. G. Averbukh, On unitary Cauchy lters on topological monoids, Topol. Algebra Appl., 1 (2013), 46-59. [2] B. G. Averbukh, On nest unitary extensions of topological monoids, Topol. Algebra Appl.,3 (2015), 1-10. [3] B. G. Averbukh, On unitary extensions and unitary completions of topological monoids, Topol. Algebra Appl., 4 (2016), 18-30. [4] B. G. Averbukh, A criterion of the existence of an embedding of a monothetic monoid into a topological group, Topol. Algebra Appl., 7(2019), 1-12. [5] R. Fric, D. C. Kent, Completion functors for Cauchy spaces, Int. J. Math. & Math. Sci. 2, No. 4 (1979), 589-604. MR 80#54042. Zbl 428.54018. [6] H. H. Keller, Die Limes-Uniformisierbarkeit der Limesräume, Math. Ann. 176 (1968), 334-341. MR 37#874. Zbl 155.50302 [7] D. C. Kent, G.D. Richardson, Regular completions of Cauchy spaces, Pacic Journal of mathematics 51 No. 2 (1974), 483-490. [8] D. C. Kent, G. D. Richardson, Cauchy completion categories, Canad. Math. Bull. 32, No. 1 (1989), 78-84. MR 90#54007. Zbl 675.54004. [9] N. Rath, Completion of a Cauchy space without the T -restriction on the space, Internat. J. Math. & Math. Sci. 24, No. 3 (2000), 163-172. [10] N. Rath, Completions of Filter Semigroups, Acta Math. Hungar. 107 (1-2) (2005), 45-54. [11] E. E. Reed, Completions of Uniform Convergence Spaces, Math. Ann. 194 (1971), 83-108. MR 45#1109. Zbl 217.19603. [12] O. Wyler, Ein Komplettierungsfunktor für uniforme Limesräume, Math. Nachr. 46 (1970), 1-12. MR 44#985. Zbl 207.52603. [13] R. Engelking, General topology. Rev. and compl. ed., Sigma Series in Pure Mathematics, 6., Berlin: Heldermann Verlag, 1989, viii + 529 pp., ISBN 3-88538-006-4, Zbl 0684.54001.

Journal

Topological Algebra and its Applications – de Gruyter

Published: Jan 1, 2022

Keywords: topological monoid; Cauchy filter; completion; 22A15; 54H10

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