Compact subsets of \(C_{\lambda,u}(X)\) (Q6550094)

Let \(X\) be a Tychonoff space, \(\lambda\) a nonempty family of subsets of \(X\) and \(C_{\lambda ,u}(X)\) the set of all real-valued continuous functions on \(X\) equipped with the topology of uniform convergence on \(\lambda\). Let \(\mathcal{F}(X)\), \(\mathcal{K}(X)\) and \(\mathcal{PS}(X)\) be the collection of finite subsets of \(X\), compact subsets of \(X\) and pseudocompact subsets of \(X\), respectively. In the paper under review, the authors give necessary and sufficient conditions for a subset to be compact in \(C_{\lambda ,u}(X)\) in each of the following cases: (1) \(X\) is a locally-\(\lambda\) space and \(\lambda\supseteq \mathcal{F}(X)\). (2) \(X\) is a hemi-\(\bar{\lambda}\) locally-\(\lambda\) space, \(\lambda \subseteq \mathcal{PS}(X)\) and \(\lambda\) is hereditarily closed with respect to closed domains. (3) \(X\) is a \(k\)-space and \(\lambda\supseteq \mathcal{K}(X)\). In the last section of the paper, the authors prove that in some classes of topological spaces \(X\) every bounded subset of \(C_{\lambda ,u}(X)\) has compact closure.