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Syntopogenous spaces

Syntopogenous spaces Acta Mathematica Academiae Scientiarum Hungaricae Tomus 25 (1--2), (1974), pp. 47--53. By M. D. ALDER (Nedlands) Introduction In Cs~sz~g's book [1], a category of objects is defined which contains the categories of topological spaces, of uniform spaces, and others. There is developed a theory of "syntopogenous spaces", structures which may be specialised by the addition of suitable extra axioms into either topologies, uniformities, proximity structures, semi-uniformities or quasi-ordoform structures. In what follows, a problem posed by CsAsz~a~ in [1] is stated and solved by what amounts to an application of the adjoint functor theorem of [2]. We change the setting of syntopogenous spaces to a more natural category theoretic one: we presuppose some familiarity with category theory and we shall occassionally cite [i] in place of a proof, otherwise this paper is substantially self contained. w 1. Preliminary definitions 1.1. REMARK. A topology on a set E determines a partial order L on the subsets of E with A L B iff A cint (B), where int (B) is the interior of B in the given topology. We start by considering order relations and aim, in particular cases, to recover the topology. 1.2. DEFINITION. L is a relation on http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Acta Mathematica Academiae Scientiarum Hungarica Springer Journals

Syntopogenous spaces

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Publisher
Springer Journals
Copyright
Copyright
Subject
Mathematics; Mathematics, general
ISSN
0001-5954
eISSN
1588-2632
DOI
10.1007/BF01901745
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Abstract

Acta Mathematica Academiae Scientiarum Hungaricae Tomus 25 (1--2), (1974), pp. 47--53. By M. D. ALDER (Nedlands) Introduction In Cs~sz~g's book [1], a category of objects is defined which contains the categories of topological spaces, of uniform spaces, and others. There is developed a theory of "syntopogenous spaces", structures which may be specialised by the addition of suitable extra axioms into either topologies, uniformities, proximity structures, semi-uniformities or quasi-ordoform structures. In what follows, a problem posed by CsAsz~a~ in [1] is stated and solved by what amounts to an application of the adjoint functor theorem of [2]. We change the setting of syntopogenous spaces to a more natural category theoretic one: we presuppose some familiarity with category theory and we shall occassionally cite [i] in place of a proof, otherwise this paper is substantially self contained. w 1. Preliminary definitions 1.1. REMARK. A topology on a set E determines a partial order L on the subsets of E with A L B iff A cint (B), where int (B) is the interior of B in the given topology. We start by considering order relations and aim, in particular cases, to recover the topology. 1.2. DEFINITION. L is a relation on

Journal

Acta Mathematica Academiae Scientiarum Hungarica – Springer Journals

Published: Jun 18, 2005

References

  • Étude comparée des topologies, proximités et uniformités par relations binaires
    Heuchenne, C.