First and second countable generators (Q2074373)

For a topological space \(X\), \(C_p(X)\) is the collection of all continuous real-valued functions defined on \(X\) with the pointwise convergence topology. The space \(C_p(X)\) is a widely studied mathematical object. A main problem in \(C_p\)-theory is to obtain topological conditions on \(C_p(X)\) that imply required topological properties on \(X\). Some of the most classic known results in this sense can be obtained through well selected subsets of \(C_p(X)\) having similar properties to those that \(C_p(X)\) should hold. In this direction, in the authors' paper [``Compact generators'', Preprint, accepted by Fundam. Math.] and [\textit{Z. Feng} et al., Acta Math. Hung. 166, No. 1, 162--178 (2022; Zbl 1499.54085)] the concept of generators was introduced. In this article, Gartside, Morgan and Yuschik define new classes of generators and produce additional results on this topic. Definition. A subset \(G\) of \(C_p(X)\) is a generator (resp., \((0,\not=0)\)-generator, or \((0,1)\)-generator) if whenever a point \(x\) is not in a closed set \(C\) of \(X\), there is a \(g \in G\) such that \(g(x) \not\in cl_{C_p(X)} g[C]\) (resp., \(g(x) \not= 0\) and \(g(C)\subseteq \{0\}\), or \(g(x)=1\) and \(g(C) \subseteq {0})\). Some results in this article are: \begin{itemize} \item[1.] A space \(X\) is separable if and only if it has a \((0, \not= 0)\)-generator containing the constant function 0, $\overline{0}$, which has a countable local base. \item[2.] If a space has a second countable \((0,\not= 0)\)-generator, then it has a second countable \((0,\not= 0)\)-generator containing $\overline{0}$. \item[3.] The following are equivalent: \begin{itemize} \item[(i)] \(X\) has a \(\sigma\)-compact cosmic generator, \item[(ii)] \(X\) has a compact second countable \((0,\not= 0)\)-generator, \item[(iii)] \(X\) embeds in \(C_p(K)\) where \(K\) is compact and second countable, and \item[(iv)] \(X\) embeds in \(C_p(Y )\) where \(Y\) is \(\sigma\)-compact and cosmic. \end{itemize} \item[4.] If \(Y\) is first countable and has a coarser second countable topology, then \(X = C_p(Y)\) has a first countable \((0,\not= 0)\)-generator containing $\overline{0}$. \item[5.] For every cardinal \(\kappa\), the compact space \(\{0,1\}^\kappa\) has a discrete \((0,1)\)-generator. Furthermore, \(\{0,1\}^\kappa\) has a first countable \((0,\not= 0)\)-generator containing $\overline{0}$ if and only if \(\kappa \leq \mathfrak{c}\). \end{itemize}