Notes on strongly Whyburn spaces. (Q2798075)

A Hausdorff space is here defined to be a \textit{strongly Whyburn space} if for any sequence \(\{A_n:n\in\omega\}\) of subsets of \(X\) and any point \(p\in X\setminus\bigcap\{\overline{\bigcup_{m\geq n} A_m}:n\in\omega\}\) there is a sequence \(\{B_n:n\in\omega\}\) of closed subsets of \(X\) such that for each \(n\in\omega\), \(B_n\subseteq A_n\) and \(\{p\}=\bigcap\{\overline{\bigcup_{m\geq n} B_m}:n\in\omega\}\). By taking \(A_n=A\) for each \(n\in\omega\), it is easy to deduce that a strongly Whyburn space is Whyburn. It is also a simple matter to show that a strongly Fréchet-Urysohn space (which has also been called a strongly bisequential space) is strongly Whyburn. What might be considered the main theorem of the paper is a characterization of the strong Whyburn property: \(X\) is strongly Whyburn if and only if \(X\times(\omega+1)\) is Whyburn. NEWLINENEWLINEThis result is analogous to the characterization of the strong Fréchet-Urysohn property given by \textit{E. A. Michael} [General Topology Appl. 2, 91--138 (1972; Zbl 0238.54009)]: A space is strongly Fréchet-Urysohn if and only if \(X\times [0,1]\) (or \(X\times(\omega+1)\)) is Fréchet-Urysohn. Among other results of the paper, there are the following: (1) If \(X\) is a \(k\)-space, then \(X\) is strongly Whyburn if and only if it is strongly Fréchet-Urysohn; (2) Every strongly Whyburn \(P\)-space is discrete, and (3) \(X\times Y\) is Whyburn for each Whyburn space \(Y\) if and only if \(X\) is discrete.