Function spaces and local properties (Q2866776)

Let \(C_{p}(X)\) denote the function space of all continuous real-valued functions on a Tychonoff space \(X\) with the topology of pointwise convergence. A space is called an \(m_1\)-space if every point in it has a closure-preserving local base. The first countable spaces are examples of such spaces. A collection \(P\) of pairs of subsets of a space is called cushioned if for every subcollection \(P'\) of \(P\) it is true that the closure of \(\bigcup \{P_1 : (P_1, P_2) \in P'\} \subset \bigcup \{\overline{P_2} : (P_1, P_2) \in P'\}\). A space \(X\) is called \(m_3\)-space if every \(x\in X\) has a cushioned local pairbase \(P_x\) (so \(P_x\) is a cushioned family of pairs and for every \(U\) containing \(x\) there is \((P_1, P_2)\) in \(P_x\) such that \(x\) in the interior of \(P_1\) and \(P_1 \subset P_2\subset U\)).NEWLINENEWLINEThe paper discusses the following questions raised by \textit{A. Dow} et al. [Topology Appl. 157, No. 3, 548--558 (2010; Zbl 1187.54021)]: (1) if \(C_{p}(X)\) is an \(m_1\), then must \(X\) be countable? (2) what about the special case when \(X\) is compact? It is shown that these questions have positive answers in various restricted cases. They hold for some properties a little weaker than the \(m_1\)-property. Necessary and sufficient conditions are obtained for \(C_{p}(X)\) to be a \(\sigma, m_1\) or \(m_3\)-space. An example of a compact uncountable space \(K\) is given with \(C_{p}(K)\) an \(m_1\)-space.