On the free set number of topological spaces and their \(G_\delta \)-modifications (Q2665213)

Let \(X\) be a topological space. A subset \(S\subset X\) is called \textit{free} in \(X\) if it admits a well-ordering that turns it into a free sequence in \(X\), where a transfinite sequence \(\{x_{\alpha}: \alpha<\eta\}\) of points of a topological space \(X\) is said to be a \textit{free sequence} in \(X\) if the closure of any initial segment of it is disjoint from the closure of the corresponding final segment. The function \(F(X)\) called the \textit{free set number} of \(X\) is defined as \[F(X)=\sup\{|S| : S\ \textmd{is free in}\ X\}.\] In this paper, the authors prove several new inequalities involving \(F(X)\) and \(F(X_{\delta})\), where \(X_{\delta}\) is the \(G_{\delta}\)-modification of \(X\): \begin{align*} &\bullet\ L(X)\leq2^{2^{F(X)}}\ \textmd{if}\ X\ \textmd{is}\ T_{2}\ \textmd{and}\ L(X)\leq2^{F(X)}\ \textmd{if}\ X\ \textmd{is}\ T_{3};\\ &\bullet\ X\leq2^{2^{F(X)\cdot\psi_{c}(X)}}\leq2^{2^{F(X)\cdot\chi(X)}}\ \textmd{for any}\ T_{3}\textmd{-space}\ X;\\ &\bullet\ \ F(X_{\delta})\leq2^{2^{2^{F(X)}}}\ \textmd{if}\ X\ \textmd{is}\ T_{2}\ \textmd{and}\ F(X_{\delta})\leq2^{2^{F(X)}}\ \textmd{if}\ X\ \textmd{is}\ T_{3}. \end{align*}