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J. Šlapal (1988)
On the axioms preserved by modifications of topologies without axioms, 24
J. Šlapal (1992)
ON CLOSURE OPERATIONS INDUCED ON GROUPOIDSDemonstratio Mathematica, 25
J. Šlapal (1988)
On modifications of topologies without axioms, 024
DEMONSTRATIO MATHEMATICAVol. XXVIIINo 11995Josef SlapalREFLECTIVE A N D COREFLECTIVE MODIFICATIONSOF THE CONSTRUCT OF TOPOLOGICAL SPACESWITHOUT AXIOMSGeneralized topologies obtained by replacing the Kuratowski closure axioms by some weaker ones occur in various branches of mathematics (seee.g. [5]) and have been studied by many mathematicians (see [3]). The mostgeneral of them — the topologies without axioms — are dealt with in thiscontribution. The upper and lower modifications of topologies without axioms with regard to the axioms O, I, M, A, U, K, B*, B, S are describedin [3]. In [4] there are determined those axioms which are preserved by theindividual modifications. By the help of [3] and [4], in the present note wesolve the problem of determining the axioms that give reflective or coreflective modifications of the construct (i.e. concrete category of structured setsand structure-compatible maps) of topological spaces without axioms. Thisis done by finding the axioms with regard to which upper (lower) modifications are reflections (coreflections).The categorical terminology used is taken from [1]. If J is a construct and(X, a) £ I an object, then by saying that (X,/3) is a reflection [coreflection]of (X, a) in a subconstruct T of 2 we mean that id^ : (X, a) —> (X,
Demonstratio Mathematica – de Gruyter
Published: Jan 1, 1995
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