On a common generalization of \(k\)-spaces and spaces with countable tightness (Q690269)

The tightness \(t(X)\) of a topological (Hausdorff) space is one of many well-known cardinal functions [see e.g. \textit{R. Hodel}, Cardinal functions, in ``Handbook of set theoretic topology'', 1-61 (1984; Zbl 0559.54003)] based on local properties. The authors generalize this function by introducing the \(k\)-tightness \(t_k (X)\) as \(t_k (X)\leq \tau\) iff for each \({M} \subseteq {X}\), \({M}\) not closed, exists a \({B}\subseteq {X}\) such that \({M}\cap {B}\) is not closed, and \({B}\) is \(\tau\)-compact. (A subset \({B}\subseteq {X}\) is \(\tau\)-compact if \({B}= \bigcup \{{B}_\alpha: \alpha\in \tau\}\) and \({B}_\alpha\) is compact in \(X\) for each \(\alpha\in \tau\).) Although several types of topological spaces have countable \(k\)-tightness (e.g. \(\sigma\)-compact spaces, \(k\)-spaces, \(p\)-spaces), in general \(k\)-tightness is different from tightness even in the case of compact spaces. \{Example: For \({X}= \beta\mathbb{N}: t_k (X)< t(X)\}\). The authors study the relation between \(t_k (X)\) and \(t(X)\). Main results are: If \(X\) is a space with only one nonisolated point, then \(t_k (X)= t(X)\). If \(X\) is a compact space then \(ht_k (X)= t(X)\) where \(ht_k (X)\) denotes the hereditary version of \(t_k (X)\).