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Ascoli’s theorem for pseudocompact spaces. (English) Zbl 1456.54006

Author’s abstract: A Tychonoff space \(X\) is called (sequentially) Ascoli if every compact subset (resp. convergent sequence) of \(C_k(X)\) is equicontinuous, where \(C_k(X)\) denotes the space of all real-valued continuous functions on \(X\) endowed with the compact-open topology. The classical Ascoli theorem states that each compact space is Ascoli. We show that a pseudocompact space \(X\) is Ascoli iff it is sequentially Ascoli iff it is selectively \(\omega\)-bounded. The class of selectively \(\omega\)-bounded spaces is studied.

MSC:

54C35 Function spaces in general topology
54D30 Compactness
54A05 Topological spaces and generalizations (closure spaces, etc.)
54B05 Subspaces in general topology
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