On minimal Hausdorff first countable spaces (Q794014)

A topological space X is called an \(E_ 1\) space if for every point x of X there is a countable collection \({\mathcal B}\) of closed neighborhoods of x such that \(\cap {\mathcal B}=\{x\}.\) By an \(\alpha\)-minimal \(E_ 1\) space the authors mean an \(E_ 1\) space X such that: for every closed subset A of X and countable cover \({\mathcal V}\) of A by open subsets of X, there is a finite subset \({\mathcal F}\) of \({\mathcal V}\) with \(A\subset cl(\cup {\mathcal F}).\) By a \(\beta\)-minimal \(E_ 1\) space the authors mean an \(E_ 1\) space X which has the following property: whenever \({\mathcal F}\) is a countable open filter base on X such that \(\cap {\mathcal F}=\cap \{cl(F):\quad F\in {\mathcal F}\},\) then \({\mathcal F}\) is a base for the family of all neighborhoods of the set \(\cap {\mathcal F}\). The authors study \(\alpha\)-minimal and \(\beta\)- minimal \(E_ 1\) spaces and obtain a number of interesting results concerning them, such as the following: An \(E_ 1\) space X is \(\beta\)- minimal, \(E_ 1\) iff every continuous mapping from X onto an \(E_ 1\) space is a closed mapping. Several examples are given.