On some properties of e-spaces

Abstract

‎An open subset of a space is said to be e-open if its closure is also open and if a space has a base consisting of e-open sets‎, ‎we call it an e-space‎. ‎In this paper we first introduce e-spaces and compare them with relative spaces such as extremally disconnected and zero-dimensional spaces‎. ‎Subspaces of e-spaces and product of e-spaces are investigated and we define the concept of e-compactness and characterize e-compact spaces via $e$-convergence of nets and filters.We introduce e-separation axioms $T_1^e-T_4^e$ and investigate the counterparts of results in the literature of topology concerning separation axioms‎. ‎It is shown that a space is a $T_3-e$-space if and only if it is zerodimensional and a space is a $T_4^e$-space if and only if it is strongly zerodimensional‎. ‎In contrast to extremally disconnected spaces whose product is not necessarily an extremally disconnected space‎, ‎we observe that any product of $e$-spaces is an $e$-space‎. ‎Also we see that the $e$-closure of a set need not be $e$-closed‎, ‎contrary to closure of a set which is closed‎.