$SG_{\delta}$-SELECTIVE SEPARABILITY

Abstract

A topological space $X$ is called $G_{\delta}$-selectively (resp., $SG_{\delta}$-selectively) separable if for every sequence $\left( D_{n}: n\in\omega\right) $ of dense $G_\delta$ subsets of $X$, one can pick finite subsets $F_{n} \subset D_{n}$ such that $\bigcup_{ n\in\omega} F_{n} $ is dense (resp., dense and $G_\delta $). In this paper we introduce and study these kinds of spaces.