On the free set number of topological spaces and their $G_\delta$-modifications

Abstract

For a topological space $X$ we propose to call a subset $S \subset X$ "free in $X$" if it admits a well-ordering that turns it into a free sequence in $X$. The well-known cardinal function $F(X)$ is then definable as $\sup\{|S| : S \text{ is free in } X\}$ and will be called the free set number of $X$. We prove several new inequalities involving $F(X)$ and $F(X_\delta)$, where $X_\delta$ is the $G_{\delta}$-modification of $X$: $\bullet$ $L(X) \le 2^{2^{F(X)}}$ if $X$ is $T_2$ and $L(X)\le 2^{F(X)}$ if $X$ is $T_3$; $\bullet$ $|X|\le 2^{2^{F(X) \cdot \psi_c(X)}} \le 2^{2^{F(X) \cdot \chi(X)}}$ for any $T_2$-space $X$; $\bullet$ $F(X_{\delta})\le 2^{2^{2^{F(X)}}}$ if $X$ is $T_2$ and $F(X_{\delta})\le 2^{2^{F(X)}}$ if $X$ is $T_3$.