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Open covers and function spaces. (English) Zbl 1119.54013

An open cover \(\mathcal U\) of a space \(X\) is said to be an \(\omega\)-cover (a \(k\)-cover) if \(X\notin\mathcal U\) and each finite (compact) subset of \(X\) is contained in a member of \(\mathcal U\). For a space \(X\), \(S_1(\mathcal K,\Omega)\) denotes the property that for each sequence \((\mathcal U_n)_{n<\infty}\) of \(k\)-covers of \(X\) there are \(U_n\in\mathcal U_n\), \(n<\infty\), such that \(\{U_n:n<\infty\}\) is an \(\omega\)-cover of \(X\). This property is characterized by a closure property of the set \(C(X)\) of continuous real-valued functions on \(X\) equipped with the compact-open and pointwise topologies. For the set \(C(X)\) with the same two topologies, the authors characterize the selective bitopological versions of the Reznichenko and Pytkeev properties, introduced by the reviewer [Acta Math. Hungar. 107, 225–233 (2005; Zbl 1082.54007)] in the context of hyperspaces.

MSC:

54C35 Function spaces in general topology
54A20 Convergence in general topology (sequences, filters, limits, convergence spaces, nets, etc.)
54D20 Noncompact covering properties (paracompact, Lindelöf, etc.)

Citations:

Zbl 1082.54007
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