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Somewhere Dense Sets and ST1-Spaces

Abstract

In this article, the main properties of a somewhere dense set [13] on topological spaces are studied and then it is used to generalize the notions of interior, closure and boundary operators. The class of somewhere dense sets contains all α-open, pre-open, semi-open, β-open and bopen sets except for the empty set. We investigate under what conditions the union of cs-dense sets and the intersection of somewhere dense sets are cs-dense and somewhere dense, respectively. A concept of ST1-space is defined and its various properties are discussed. Theorem (4.1) and Corollary (4.14) give the answer for why we do not define ST0-spaces, ST 1 2 -spaces, ST2-spaces and ST2 1 2 -spaces. Also, we point out that the product of ST1-spaces is always an ST1-space and present some examples to illustrate the main results.

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