Small diagonal of \(X\) and calibers of \(C_{p}(X)\)
The paper under review revisits the connections between smallness of the diagonal of a topological space \(X\) and the calibre of \(C_p(X)\), the space of all real-valued continuous functions with the topology of pointwise convergence. Let \(\kappa\) be an uncountable cardinal number (for example, a successor cardinal number such as \(\aleph_1\)) and let \(Z\) be a topological space. The number \(\kappa\) is called a calibre of \(Z\) (in American English: caliber) whenever every family \(\mathcal U\) with cardinality \(\kappa\) comprising non-empty open subsets of \(Z\) contains a subfamily \(\mathcal{V}\) with the same cardinality and non-empty intersection. Moreover, \(Z\) has \(\kappa\) small diagonal whenever any set \(A\) with cardinality \(\kappa\) contained in the complement of the diagonal \(\Delta = \{(z,z): z\in Z\}\) contains a subset \(B\) with the same cardinality such that \(\overline{B}\cap \Delta=\varnothing\). As proved by \textit{A. V. Arkhangel'skij} and the author [Topology Appl. 23, 65--73 (1986; Zbl 0591.54023)], for a compact space \(X\) and an uncountable cardinal \(\kappa\), \(\kappa\) is a calibre of \(C_p(X)\) if and only if \(X\) has \(\kappa\)-small diagonal. After over 30 years, this result is extended to all Lindelöf spaces. Interestingly, a pseudo-compact space \(X\) is constructed such that \(\omega_1\) is a calibre of \(C_p(X)\) neverthless it is not a calibre of the \(C_p\)-space over the one-point compactification of \(X\) (which happens to be the Čech-Stone compactification of \(X\)). The paper is concluded with a list of eleven open problems. Let us hope they do not need another 30 years to be solved.
scientific article; zbMATH DE number 6986549
scientific article