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REFLECTIVE AND COREFLECTIVE MODIFICATIONS OF THE CONSTRUCT OF TOPOLOGICAL SPACES WITHOUT AXIOMS

REFLECTIVE AND COREFLECTIVE MODIFICATIONS OF THE CONSTRUCT OF TOPOLOGICAL SPACES WITHOUT AXIOMS 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, http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Demonstratio Mathematica de Gruyter

REFLECTIVE AND COREFLECTIVE MODIFICATIONS OF THE CONSTRUCT OF TOPOLOGICAL SPACES WITHOUT AXIOMS

Demonstratio Mathematica , Volume 28 (1): 8 – Jan 1, 1995

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References (3)

Publisher
de Gruyter
Copyright
© by Josef Šlapal
ISSN
0420-1213
eISSN
2391-4661
DOI
10.1515/dema-1995-0102
Publisher site
See Article on Publisher Site

Abstract

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,

Journal

Demonstratio Mathematicade Gruyter

Published: Jan 1, 1995

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