Topologies between compact and uniform convergence on function spaces
The authors consider two new topologies on the set \(C(X)\) of all continuous real-valued functions on a Tikhonov space \(X\). The first of those topologies is defined in the following rather general manner: Let \(\alpha\) be a family of subsets of \(X\) which is directed by inclusion. For \(f\in C(X)\), \(A\in \alpha\) and \(\varepsilon>0\), let \(\langle f,A, \varepsilon\rangle= \{g\in C(X)\); \(| f(x)- g(x)|< \varepsilon\) for all \(x\in a\}\). The topology of uniform convergence on \(\alpha\) is the one defined by means of the neighbourhood base \(\{\langle f,A, \varepsilon\rangle\): \(A\in\alpha\), \(\varepsilon>0\}\) at each \(f\in C(X)\). In the case when \(\alpha\) is the set of all \(\sigma\)-compact subsets of \(X\), the resulting topology is called the topology of uniform convergence on \(\sigma\)-compact sets and is denoted by \(C_{\sigma, u}(X)\). The other topology is defined as follows: For each subset \(A\) of \(X\) and each open set \(V\) of the reals, let \([A,V]= \{f\in C(X)\): \(\text{cl}(f(A))\subset V\}\). Then the collection of all sets \([A,B]\), where \(A\) is a \(\sigma\)-compact subset of \(X\) and \(B\) is a bounded open interval, is a subbasis for the \(\sigma\)-compact-open topology \(C_ \sigma(X)\) on \(C(X)\). The authors investigate \(C_{\sigma,u}(X)\) and \(C_ \sigma(X)\) obtaining many interesting results. The following open problem is posed: When is \(C_ \sigma(X)\) metrizable?
scientific article; zbMATH DE number 130459