SOME REMARKS ON MONOTONICALLY STAR COUNTABLE SPACES

Abstract

A topological space $X$ is monotonically star countable if for every open cover $\mathcal U$ of $X$ we can assign a subspace$s(\mathcal U)\subseteq X$, called the kernel, such that $s(\mathcalU)$ is a countable subset of $X$, and $st(s(\mathcal U),\mathcalU)=X$, and if $\mathcal V$ refines $\mathcal U$, then $s(\mathcalU)\subseteq s(\mathcal V)$, where $st(s(\mathcal U),\mathcalU)=\bigcup\{U\in\mathcal U:U\cap s(\mathcal U)\neq\emptyset\}.$In this paper we study the relation betweenmonotonically star countable spaces and related spaces, and we also study topological properties of monotonically star countablespaces.