The metrisability of precompact sets

Abstract

A uniform space is trans-separable if every uniform cover has a countable subcover. We show that a uniform space is trans-separable if it contains a suitable family of precompact sets. Applying this result to locally convex spaces, we are able to deduce that the precompact subsets of a wide class of spaces are metrisable. The proof of our main Theorem is based on a cardinality argument, and is reminiscent of the classical Bolzano-Weierstrass Theorem.A uniform space X is said to be trans-separable if every uniform cover of X admits a countable subcover. Expressed in terms of entourages or surroundings, this means that for each entourage H in X x X, there is a countable subset C of X satisfying H[C\ = X. This condition is strictly weaker than the usual topological notion of separability: every separable uniform space is trans-separable, but not conversely. The class of transseparable spaces forms a reflective subcategory of the category of uniform spaces; it is closed under the taking of completions, subspaces, products, uniformly continuous images and projective limits. Every Lindelof uniform space is trans-separable, but again the converse does not hold.The concept of trans-separability is sometimes encountered in the context of topological vector spaces. Pfister [6] has shown that a locally convex space is trans-separable if and only if the equicontinuous subsets of its dual are weak* metrisable. (For simplicity, we shall assume that all our spaces are Hausdorff.) As a consequence, many results in functional analysis which are normally stated in terms of separable spaces actually only require the weaker condition of trans-separability.Trans-separability appears under many different names in the literature. While workers in the field of uniform spaces tend to speak simply of separable spaces (see [3] and [4]), those who study locally convex spaces favour the phrase "separable by seminorm". Pfister investigated topological vector spaces which were "of countable type", whereas Drewnowski [2] coined the word "trans-separable" while working with topological abelian groups.Pfister used the idea of trans-separability to show that in a DF space precompact sets are metrisable. (For the definition of a DF space, and other terms from the theory

Necessary and sufficient conditions for precompact sets to be metrisable

“…Since, according to (B), every locally convex space covered by an ordered family of precompact sets is trans-separable, Theorem 3 includes Robertson's [16,Corollary 2]. Likewise, Cascales-Orihuela's metrisation theorem (A) follows immediately from the previous theorem and the result (B), as the following corollary shows.…”

“…Recall that a locally convex space E is trans-separable if for every absolutely convex neighbourhood of zero U in E there exists a countable subset N of E such that E -N + U, [12], see also [14,15,16], where this notion has been studied (also) in uniform spaces. Every separable and every Lindelof locally convex spaces are transseparable, and in the class of metrisable locally convex spaces the trans-separability and separability are equivalent notions.…”

“…Indeed, since a locally complete Mackey space E is /f-analytic if and only if E is covered by an ordered family 8 J.C. Ferrando, J. Kakol and M. Lopez Pellicer [2] of compact sets [4 Result (B) is nicely applied in [16] to show that if the topological dual E' of a locally convex space E endowed with the topology r p of uniform convergence on the precompact sets of E satisfies the assumption stated in (B), every precompact set in E is metrisable. Nevertheless, the converse statement fails: If X is an uncountable completely regular Hausdorff P-space, the space C P {X) of all real-valued continuous functions on X equipped with the pointwise topology is barrelled [3] and its dual C P (X)' with the weak* topology cannot be covered by sets as stated in (B) (so the same property holds for (Cp(X)',T p )).…”

Trans-Separability in Spaces of Continuous Vector-Valued Functions

“…A uniform space L is called trans-separable if every uniform cover of L admits a countable subcover ( [7], [6]). This notion has occured to be useful in the work of ( [14], [1], [15]) while studying the metrizability of precompact sets in locally convex spaces (see also [9]). Drewnowski [2] had actually coined the word "trans-separable" and it has been further used by Robertson [15].…”

“…This notion has occured to be useful in the work of ( [14], [1], [15]) while studying the metrizability of precompact sets in locally convex spaces (see also [9]). Drewnowski [2] had actually coined the word "trans-separable" and it has been further used by Robertson [15]. Recently, in [10], the author has introduced a generalized notion of separability in the TVS setting and characterized it for vectorvalued function spaces endowed with the strict and compact-open topologies.…”