Some remarks on set tightness (Q1098128)

The author studies a cardinal function \(t_ S\) called set-tightness which is defined for a topological space X as the smallest cardinal \({\mathfrak m}\) such that for any closed subset A of X and any \(x\in \bar A- A\) there exists a family \({\mathcal B}\) of subsets of A with card \({\mathcal B}\leq {\mathfrak m}\) such that \(x\in \overline{\cup_{B\in {\mathcal B}}B}\) but \(x\not\in \bar B\) for any \(B\in {\mathcal B}\). Some necessary conditions for the equality of set-tightness and tightness are proved and upper estimates of a set-tightness for some pseudo-radial regular spaces are given. Further the author studies the relationship between set-tightness and quasi-character \(\lambda\) defined for a topological space X as the smallest cardinal \({\mathfrak m}\) such that for any open subset U of X there exists a family \({\mathcal F}\) of closed subsets of X with card \({\mathcal F}\leq {\mathfrak m}\) such that \(\cup_{F\in {\mathcal F}}F\subseteq U\subseteq \overline{\cup_{F\in {\mathcal F}}F}\).