Abstract
Via co-compact open sets we introduce co-T2 as a new topological property. We show that this class of topological spaces strictly contains the class of Hausdorff topological spaces. Using compact sets, we characterize co-T2 which forms a symmetry. We show that co-T2 propoerty is preserved by continuous closed injective functions. We show that a closed subspace of a co-T2 topological space is co-T2. We introduce co-regularity as a weaker form of regularity, s-regularity as a stronger form of regularity and co-normality as a weaker form of normality. We obtain several characterizations, implications, and examples regarding co-regularity, s-regularity and co-normality. Moreover, we give several preservation theorems under slightly coc-continuous functions.
Highlights
Introduction and PreliminariesDefining a new type of generalized open sets and utilizing it to define new topological concepts is a very hot research topic [1,2,3,4,5,6,7,8,9,10,11]
As a new type of generalized open sets, Al-Ghour and Samarah in [12] defined coc-open sets as follows: A subset A of a topological space ( X, τ ) is called coc-open set if A is a union of sets of the form V − C, where V ∈ τ and C is a compact subset of X
Proved that the family of all coc-open sets of a topological space ( X, τ ) forms a topology on X finer than τ, and via this class of sets they obtained a decomposition theorem of continuity
Summary
Defining a new type of generalized open sets and utilizing it to define new topological concepts is a very hot research topic [1,2,3,4,5,6,7,8,9,10,11]. As a new type of generalized open sets, Al-Ghour and Samarah in [12] defined coc-open sets as follows: A subset A of a topological space ( X, τ ) is called coc-open set if A is a union of sets of the form V − C, where V ∈ τ and C is a compact subset of X. Proved that the family of all coc-open sets of a topological space ( X, τ ) forms a topology on X finer than τ, and via this class of sets they obtained a decomposition theorem of continuity. New classes of functions were introduced, and in [19] the suthors generalized co-compact open sets. We use coc-open sets to define and investigate new separation axioms and new class of functions. {U − K : U ∈ τ and K is compact in ( X, τ )} forms a base for τ k
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