Seperation Axioms in the Structural Topology
Subject Areas : StatisticsMohammad Zaher Kazemi Baneh 1 , Seyed Naser Hosseini 2
1 - Department of Mathematics, Faculty of Sciences, University of Kurdistan, Sanandaj, Iran.
2 - Department of Pure Mathematics, Faculty of Mathematics and Computer,Shahid Bahonar Univercity of Kerman, Kerman, Iran
Keywords: closed subobject, open subobject, seperation axioms, structural topology, topological structure.,
Abstract :
The notion of structural topology is introduced in [3]. There it is shown that standard topology as well as several fuzzy topologies fall within this framework. The category STop of structural topological spaces is defined in [5], where it is shown that under certain conditions, the category has equalizers, terminal objects, binary products and (finite) limits. Here we define singleton subject, open and closed subobjects. So we introduce separation axioms as T0, T1 and Hausdorffness, relative to a given singleton function, for an object in Stop. We investigate some equivalents of Hausdorffness and we prove under certain conditions the T0 property is equivalent to closedness of singletons and every singleton is the meet of a class of open subobjects. At the end we give several examples as standard topological spaces category, fuzzy topological spaces category, small complete lattice in which binary meet distributes over arbitrary join as a complete category and we investigate separation axioms on the objects of those categories.
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