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.


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.)


Zbl 1082.54007
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