Ascoli and sequentially Ascoli spaces

Abstract

A Tychonoff space X is called (sequentially) Ascoli if every compact subset (resp. convergent sequence) of Ck(X) is evenly continuous, where Ck(X) denotes the space of all real-valued continuous functions on X endowed with the compact-open topology. Various properties of (sequentially) Ascoli spaces are studied, and we give several characterizations of sequentially Ascoli spaces. Strengthening a result of Arhangel'skii we show that a hereditary Ascoli space is Fréchet–Urysohn. A locally compact abelian group G with the Bohr topology is sequentially Ascoli iff G is compact. If X is totally countably compact or near sequentially compact then it is a sequentially Ascoli space. The product of a locally compact space and an Ascoli space is Ascoli. If additionally X is a μ-space, then X is locally compact iff the product of X with any Ascoli space is an Ascoli space. Extending one of the main results of [18] and [16] we show that Cp(X) is sequentially Ascoli iff X has the property (κ). We give a necessary condition on X for which the space Ck(X) is sequentially Ascoli. For every metrizable abelian group Y, Y-Tychonoff space X, and nonzero countable ordinal α, the space Bα(X,Y) of Baire-α functions from X to Y is κ-Fréchet–Urysohn and hence Ascoli.