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Countable product of function spaces having \(p\)-Fréchet-Urysohn like properties. (English) Zbl 0888.54019
This paper centers around Fréchet-Urysohn like properties on spaces of continuous functions from \(X\) to \(Y\), where the topology is in general that of uniform convergence on sets in a given network \(\mathcal A\) on \(X\) relative to a given uniformity \(\mathcal U\) on \(Y\) (such spaces are denoted by \(C_{\mathcal A,\mathcal U}(X,Y)\)). For example, one property studied in the paper is FU(\(p\))-space for \(p\in\omega^*\), which means that every limit point of a subset is a \(p\)-limit of some sequence in the subset. It is shown that when \(Y\) is metrizable and has a non-trivial path, \(C_{\mathcal A,\mathcal U}(X,Y)\) is an FU(\(p\))-space if and only if \(X\) has property \(\mathcal A\gamma_p\) (the natural generalization of having property \(\gamma\)). Also countable products of such function spaces are considered, and a corollary of a general result is that, under the topology of pointwise convergence, \(C_p(X)\) is an FU(\(p\))-space if and only if \(\prod_{n<\omega}C_p(X^n)\) is an FU(\(p\))-space. Other related topics include: topological game characterization of these properties; the dual space of \(C_p(X)\) and free topological groups generated by \(X\); the \(\gamma_p\) property for \(p\) such that \(R\) does not have \(\gamma_p\); and compactifications of \(X\) that retain the Fréchet-Urysohn property on the function space.
MSC:
54C35 Function spaces in general topology
54D55 Sequential spaces
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