The Ascoli property for function spaces

摘要

The paper deals with Ascoli spaces C-p(X) and C-k(X) over Tychonoff spaces X. The class of Ascoli spaces X, i.e. spaces X for which any compact subset K of C-k(X) is evenly continuous, essentially includes the class of k(R)-spaces. First we prove that if C-p(X) is Ascoli, then it is k-Frechet-Urysohn. If X is cosmic, then C-p(X) is Ascoli iff it is k-Frechet-Urysohn. This leads to the following extension of a result of Morishita: If for a Cech-complete space X the space C-p(X) is Ascoli, then X is scattered. If X is scattered and stratifiable, then C-p(X) is an Ascoli space. Consequently: (a) If X is a complete metrizable space, then C-p(X) is Ascoli iff X is scattered. (b) If X is a Cech-complete Lindelof space, then C-p(X) is Ascoli iff X is scattered iff C-p(X) is Frechet-Urysohn. Moreover, we prove that for a paracompact space X of point-countable type the following conditions are equivalent: (i) X is locally compact. (ii) C-k(X) is a k(R)-space. (iii) C-k(X) is an Ascoli space. The Ascoli spaces C-k(X, I) are also studied. (C) 2016 Elsevier B.V. All rights reserved.