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Topological-algebraic properties of function spaces with set-open topologies. (English) Zbl 1243.54037
Continuing his work on the coincidence of different topologies on \(C(X)\), the space of all continuous real-valued functions on a Tychonoff space \(X\) in [Topol. Proc. 37, 205–217 (2011; Zbl 1227.54021)], the author investigates the topological-algebraic properties of \(C_\lambda(X)\), the set-open topology for which the sets [F,U]=\(\{f \in C(X): f(F) \in U\}\) for \(F \in \lambda\), \(\lambda\) is a family of non-empty subsets of \(X\) and \(U\) open in \(\mathbb{R}\), form a subbase. When \(\lambda '\) consists of all \(C\)-compact subsets of every set of \(\lambda\), (i.e., \(f(A)\) is compact in \(\mathbb{R}\) for every continuous function \(f\) on \(X\), and \(A \in \lambda\)), the topology similarly generated is denoted by \(C_{\lambda '} (X)\). Here, \(\lambda\) is a \(\pi\)-net (i.e., for any open set \(U\) in \(X\), there is \(A \in \lambda\) such that \(A \subset U\)) which is equivalent to the space \(C_{\lambda}(X)\) being Hausdorff.
Contained in a series of equivalent statements, one main result is that \(C_{\lambda}(X)\) is a topological vector space (a topological group) if and only if \(\lambda\) is a family of \(C\)-compact sets and \(C_{\lambda}(X) = C_{\lambda '}(X)\). In particular, if \(C_{\lambda}(X)\) is a topological group, then the set-open topology coincides with the topology of uniform convergence on a family \(\lambda\) generated by the base of sets of the form \(\{g \in C(x): sup_{x \in F}\{|g(x)-f(x)|<\epsilon\}\) where \(F \in \lambda\) \(\epsilon >0\). Another result uses a cardinal invariant of a topological group \(G\): \(G\) is \(\omega\)-narrow if for every neighborhood \(U\) of the identity in \(G\), there is a subset \(S\) of \(G\) such that \(|S| \leq \omega\) and \(G = \{s+u: s \in S, u \in U\}\). The author shows that a topological group \(C_{\lambda}(X)\) is \(\omega\)-narrow if and only if \(\lambda\) is a family of metrizable compact subsets of \(X\).

54C40 Algebraic properties of function spaces in general topology
54C35 Function spaces in general topology
54D60 Realcompactness and realcompactification
54H11 Topological groups (topological aspects)
46E10 Topological linear spaces of continuous, differentiable or analytic functions
54A10 Several topologies on one set (change of topology, comparison of topologies, lattices of topologies)
Full Text: DOI
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