Tight points and countable fan-tightness
A family \(\mathcal E\) of subsets of a space \(X\) clusters at a point \(p \in X\) if for every neighbourhood \(U\) of \(p\) there is a member \(E\) in \(\mathcal E\) such that the set \(E \cap U\) is infinite. A point \(p\) is a tight point of \(X\) if for every family \(\mathcal E\) of subsets of \(X\) which clusters at \(p\) there is a countable subfamily \(\mathcal F\) of \(\mathcal E\) which clusters at \(p\). Recall that a space \(X\) has countable fan-tightness if for every countable family \(\{A_n:n \in \omega\}\) of subsets of \(X\) with \(x \in \bigcap_{n \in \omega}\overline{A_n}\) there are finite sets \(B_n \subset A_n\), \(n \in \omega\), such that \(x \in \overline{\bigcup_{n \in \omega}B_n}\). It is shown that a regular countably compact space \(X\) has countable tightness iff \(X\) has countable fan-tightness iff every point of \(X\) is tight. For a countable space \(X\) the product of \(X\) with the sequential fan \(S_c\) (obtained by identifying the limit points of the topological sum of continuum many convergent sequences) has countable tightness if every point of \(X\) is tight. A corollary of this result answers a question of Arkhangel'skij and the first author: If \(X\) is a space such that \(X \times S_c\) has countable tightness, then \(X\) has countable fan-tightness.
scientific article; zbMATH DE number 991202