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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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