Somewhere Dense Sets and ST1-Spaces
Keywords:
Somewhere dense set, Hyperconnected space, Product spaces, ST1-spaces and topological spacesAbstract
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) andCorollary (4.14) give the answer for why we do not define ST0-spaces,ST 12-spaces, ST2-spaces and ST212-spaces. Also, we point out that theproduct of ST1-spaces is always an ST1-space and present some examples to illustrate the main results.
