A note on monotonically star $\sigma$-compact spaces

Abstract

A space $X$ is {\it monotonically star $\sigma$-compact} if one assigns to each open cover $\mathcal U$ of $X$ a subspace $s(\mathcal U)\subseteq X$, called a kernel, such that $s(\mathcal U)$ is a $\sigma$-compact subset of $X$, and $st(s(\mathcal U),\mathcal U)=X$, and if $\mathcal V$ refines $\mathcal U$ then $s(\mathcal U)\subseteq s(\mathcal V)$, where $st(s(\mathcal U),\mathcal U)=\bigcup\{U\in \mathcal U:U\cap s(\mathcal U)\neq\emptyset\}.$ In this paper, we investigate the relationship between monotonically star $\sigma$-compact spaces and related spaces, and also study topological properties of monotonically star $\sigma$-compact spaces.