The Ascoli property for function spaces

Abstract

Abstract. The paper deals with Ascoli spaces Cp(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 Cp(X) is Ascoli, then it is κ-Fréchet-Urysohn. If X is cosmic, then Cp(X) is Ascoli iff it is κ-Fréchet-Urysohn. This leads to the following extension of a result of Morishita: If for aČech-complete space X the space Cp(X) is Ascoli, then X is scattered. If X is scattered and stratifiable, then Cp(X) is an Ascoli space. Consequently: (a) If X is a complete metrizable space, then Cp(X) is Ascoli iff X is scattered. (b) If X is aČech-complete Lindelöf space, then Cp(X) is Ascoli iff X is scattered iff Cp(X) is Fréchet-Urysohn. Moreover, we prove that for a paracompact space X of point-countable type the following conditions are equivalent:is an Ascoli space. The Asoli spaces C k (X, I) are also studied.

On the Ascoli property for locally convex spaces

“…Note that X ′ = X ′ . So (X, σ(X, X ′ )) is a dense subspace of (X, σ(X, X ′ )), and hence (X, σ(X, X ′ )) is an Ascoli space by Lemma 2.7 of [12]. Therefore the topology τ of X coincides with σ(X, X ′ ) by Theorem 1.5.…”

“…and none of these implications is reversible. The Ascoli property for function spaces has been studied recently in [3,4,10,12,13,16]. Let us mention the following Theorem 1.1.…”

Topological properties of spaces of Baire functions

“…In [28, Theorem 2.1], Sakai showed that C p (X) is κ-Fréchet-Urysohn if and only if X has the property (κ). In [11] we proved that if C p (X) is Ascoli, then it is κ-Fréchet-Urysohn. These results and Theorem 2.5 immediately imply Corollary 2.6.…”

“…Clearly, σ(z) is a dense subspace of X. Proposition 2.6 of [11] states that σ(z) is Fréchet-Urysohn. Thus, by Theorem 2.1, X is κ-Fréchet-Urysohn.…”

“…Proof. If C p (X) is Ascoli, then X is scattered by Theorem 1.3 of [11]. Conversely, if X is scattered, then, by Corollary 3.8 of [28], X has the property (κ).…”