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 $s(\text{mathcalU})$ is a countable subset of $X$, and $st(s(\mathcal{U}),\text{mathcalU})=X$, and if $\mathcal{V}$ refines $\mathcal{U}$, then $s(\text{mathcalU})\subseteq s(\mathcal{V})$, where $st(s(\mathcal{U}),\text{mathcalU})=\bigcup \{U\in \mathcal{U}:U\cap s(\mathcal{U})\ne \mathrm{\varnothing }\}.$In this paper we study the relation betweenmonotonically star countable spaces and related spaces, and we also study topological properties of monotonically star countablespaces.