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 $T_3$.

arXiv comment6 pages