Generalization of ω-closed sets via operators and ideals

Abstract

A new type of closed sets in a topological space, called ωI,γclosed sets, is introduced and studied. Also we studied and characterized the class of spaces having (I, γ)-tightness.

A unified theory of weakly g-closed sets and weakly g-continuous functions

“…The complement of an ω I,γ -closed set is called ω I,γ -open. Each γ-open set is an ω I,γ -open set [3].…”

“…In a similar form as for the closure and the interior of a subset A in a topological space, we can define [3]:…”

“…Since f is an ω I,γ -continuous function and cl(U ), cl(Y \ U ) are closed subsets in Y , then f −1 (cl(U )) and f −1 (cl(Y \ U )) are ω I,γ -closed sets in X. Using that the intersection of ω I,γ -closed sets is an ω I,γ -closed set [3], then f −1 (F r(U )) is a ω I,γ -closed set in X. This shows that f is a coweakly ω I,γ -continuous function.…”

“…In 2012, Carpintero et al [3], introduced the concept of ω I,γ -closed sets in terms of operators on topological spaces as a generalization of the concept given by Arhangel'skiȋ [1] and Ekici et al in [6]. The purpose of this article is to introduce and to study a new class of functions called ω I,γ -continuous and weakly ω I,γ -continuous in terms of the ω I,γ -closed sets.…”

Omega open sets in generalized topological spaces

“…The family of all ω-open subsets of X forms a topology on X, denoted by τ ω . Many topological concepts and results related to ω-closed and ω-open sets appeared in [1,2,5,6,7,8,10,11,20,29,31] and in the references therein. In 2002, Császár [12] defined generalized topological spaces as follows: the pair (X, µ) is a generalized topological space if X is a nonempty set and µ is a collection of subsets of X such that ∅ ∈ µ and µ is closed under arbitrary unions.…”