Coincidence of function space topologies
Given topological spaces \(X\) and \(Y\), let \(C(X,Y)\) be the set of all continuous functions from \(X\) to \(Y\). It is known that \(C(X,Y)\) can be equipped with various topologies such as the compact-open topology \(\tau_{co}\), the Isbell topology \(\tau_{is}\), and the topology \(\tau(c)\) of continuous convergence. In general, we have \(\tau_{co} \subseteq \tau_{is} \subseteq \tau(c)\) on \(C(X,Y)\). If \(\tau_{co}=\tau_{is}\) on \(C(X,Y)\), then \(X\) is called \textit{\(Y\)-consonant}; if \(\tau_{is} = \tau(c)\) on \(C(X,Y)\), then \(X\) is called \textit{\(Y\)-concordant}; and finally if \(\tau_{co} =\tau(c)\) on \(C(X,Y)\), then \(X\) is called \textit{\(Y\)-harmonic}. In this paper, the author studies when these topologies coincide on \(C(X,{\mathbb R})\), and investigates \(\mathbb R\)-concordant and \(\mathbb R\)-harmonic spaces. It is shown that a metric space is locally compact if and only if it is \(\mathbb R\)-concordant. It is also shown that the countable sequential fan is a \(\mathbb R\)-harmonic and non-locally compact space, which answers part of Problem 13 in [\textit{E. Pearl}, Open problems in topology. II. (pp. 15-22), Amsterdam: Elsevier. (2007; Zbl 1158.54300)]. Assuming \(\omega_1 <\mathfrak b\), the author constructs a \(\mathbb R\)-concordant non-consonant (thus, non-\(\mathbb R\)-harmonic) space, which provides a consistent answer to Problem 15 in the same reference mentioned previously.
scientific article; zbMATH DE number 5647954
scientific article