Pointwise Compactness in Spaces of Continuous Functions

Abstract

In this paper we describe a class of topological spaces X such that C p (X), the space of continuous functions on A'endowed with the topology of pointwise convergence, is an angelic space. This class contains the topological spaces with a dense and countably determined subspace; in particular the topological spaces which are ^-analytic in the sense of G. Choquet. Our results include previous ones of A. Grothendieck, J. L. Kelley and I. Namioka, J. D. Pryce, R. Haydon, M. De Wilde, K. Floret and M. Talagrand. As a consequence we obtain an improvement of the Eberlein-Smulian theorem in the theory of locally convex spaces. This result allows us to deduce, for instance, that (LF)-spaces and dual metric spaces, in particular (Z)F)-spaces of Grothendieck, are weakly angelic. In this way the answer to a question posed by K. Floret about the weak angelic character of (L/)-spaces is given.

James boundaries and σ-fragmented selectors

“…The equality in (i) follows from Lemma 3 if we bear in mind that T ∪ {y * } is countably K-determined for every y * ∈ X * and therefore the space C p (T ∪{y * }) is angelic [41]. If (T, w) is countably K-determined (resp.…”

“…The paper [50] is a milestone in the study of Banach spaces which are countably K-determined when endowed with their weak topologies. The main result in [41] states that if T is a countably K-determined space then C p (T ) is angelic.…”

“…A topological space T is said to be angelic if, whenever A is a relatively countably compact subset of T , its closure A is compact and each element of A is a limit of a sequence in A; good references for angelic spaces are [15] and [41]. C p (T ) stands for the space of real continuous functions endowed with the topology of pointwise convergence on T .…”

The Lindelöf property and σ-fragmentability

“…(IV) A topological space T is said to be angelic if, whenever C is a relatively countably compact subset of T , its closure C is compact and each element of C is a limit of a sequence in C. It is known that (C(X), τ p ) is angelic whenever X is K-analytic (see [22] …”

A quantitative approach to weak compactness in Fréchet spaces and spacesC(X)

“…The approximation Theorem 7 (extending [2, Theorem 3.2] to Fréchet spaces) is the quantitative version of the weak angelicity of a Fréchet space. Theorem 14 and Corollary 15 provide a quantitative version of Orihuela's angelic theorem [18,Theorem 3] showing that ck(H) ≤d(H, C(X, Z)) ≤ 17ck(H), where H ⊂ Z X is relatively compact and C(X, Z) is the space of Z-valued continuous functions for webcompact spaces X and separable metric space Z. If X is web-compact and normal and Z := R, then ck(H) ≤d(H, C(X)) ≤ 12ck(H).…”