Abstract
We introduce and investigate the concepts of θ ω -limit points and θ ω -interior points, and we use them to introduce two new topological operators. For a subset B of a topological space Y , σ , denote the set of all limit points of B (resp. θ -limit points of B , θ ω -limit points of B , interior points of B , θ -interior points of B , and θ ω -interior points of B ) by D B (resp. D θ B , D θ ω B , Int B , Int θ B , and Int θ ω B ). Several results regarding the two new topological operators are given. In particular, we show that D θ ω B lies strictly between D B and D θ B and Int θ ω B lies strictly between Int θ B and Int B . We show that D B = D θ ω B (resp. Cl θ B = Cl θ ω B and D B = D θ ω B = D θ B ) for locally countable topological spaces (resp. antilocally countable topological spaces and regular topological spaces). In addition to these, we introduce several product theorems concerning metacompactness.
Highlights
In 1943, Fomin [1] introduced the notion of θ-continuity
Several topological properties are not finitely productive, such as paracompactness, strong paracompactness, Lindelofness, and metacompactness. e area of research regarding the problem “What conditions on (Y, σ) and (Z, δ) to insure that their product has property P”is still hot[38,39,40,41,42,43,44,45]. e second goal of this paper is to introduce several product theorems concerning metacompactness
Let (Y, σ) and (Z, δ) be TSs and let B⊆C⊆Y with C as nonempty. en, B is called ω-open set in (Y, σ) [27] if for each y ∈ B, there is M ∈ σ and a countable set F⊆Y such that y ∈ M − F⊆B. e relative topology on C is denoted by σC, and the product topology on Y × Z is denoted by σ × δ. e closure of B in
Summary
In 1943, Fomin [1] introduced the notion of θ-continuity. For the purpose of studying the important class of H-closed spaces in terms of arbitrary filterbases, the notions of θ-open subsets, θ-closed subsets, and θ-closure were introduced by Velicko [2] in 1966, in which he showed that the family of θ-open sets in a topological space (Y, σ) forms a topology on Y denoted by σθ (see [3]). e work of Velicko is continued by [3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26] and others. In 2017, Al Ghour and Irshidat [37] introduced θω-open subsets, θω-closed subsets, and θω-closure utilizing the topological spaces (Y, σθ) and (Y, σω). It is proved in [37] that σθω forms a topology on Y which lies between σθ and σ, and that σθω σ if and only if (Y, σ) is ω-regular. Judging from the importance of limit points in mathematical analysis, introducing a new limit point notion in any topological structure is still a hot area of research. Several topological properties are not finitely productive, such as paracompactness, strong paracompactness, Lindelofness, and metacompactness. e area of research regarding the problem “What conditions on (Y, σ) and (Z, δ) to insure that their product has property P”is still hot[38,39,40,41,42,43,44,45]. e second goal of this paper is to introduce several product theorems concerning metacompactness
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