Compact coverings for Baire locally convex spaces

Abstract

Very recently Tkachuk has proved that for a completely regular Hausdorff space X the space C p (X) of continuous real-valued functions on X with the pointwise topology is metrizable, complete and separable iff C p (X) is Baire (i.e. of the second Baire category) and is covered by a family {K α : α ∈ N N } of compact sets such that K α ⊂ K β if α β. Our general result, which extends some results of De Wilde, Sunyach and Valdivia, states that a locally convex space E is separable metrizable and complete iff E is Baire and is covered by an ordered family {K α : α ∈ N N } of relatively countably compact sets. Consequently every Baire locally convex space which is quasi-Suslin is separable metrizable and complete.

On realcompact topological vector spaces

“…Since E has a σ (E, E )-compact resolution and the original topology of E has a basis of neighbourhoods consisting of σ (E, E )-closed sets, then the space E admits a complete resolution as well. But every metrizable Baire lcs which has a complete resolution is complete, see [34,Theorem] and its refined version in [15, Theorem 3.5 and Corollary 3.6].…”

Web-compact spaces, Fréchet-Urysohn groups and a Suslin closed graph theorem

“…Any K-analytic space is dominated by irrationals [31] (see also [5,32]) and every metrizable space dominated by irrationals is separable by [32,Proposition 2.1]. Any Polish space is strongly dominated by irrationals; see [21,32] ( [23] for recent results concerning Banach spaces strongly dominated by irrationals for its weak topology). In [32, Example 3.5] Tkachuk provided an example of a locally compact, countably compact non-compact (hence not Lindelöf) web-compact space X which is strongly dominated by irrationals.…”

Resolutions of topological linear spaces and continuity of linear maps

“…(Actually, not only compact subsets will enter the scene.) We are primarily interested in a more specific question that has been raised in a recent paper by J. Kakol and M. López Pellicer [8,Problem]:…”

“…As it was explained in [8], if the answer were 'yes,' the kernel of such a functional f would be a non-Baire dense hyperplane in X, thus providing an alternative proof, possibly without using the Continuum Hypothesis, for a deep result of Arias de Reyna (1980). Furthermore, it was already shown in [8,Proposition 5] that the answer is 'no' if the sets K α are additionally required to be absolutely convex.…”

“…Furthermore, it was already shown in [8,Proposition 5] that the answer is 'no' if the sets K α are additionally required to be absolutely convex. It was, therefore, reasonable to wonder if, after replacing the sets K α with their closed absolutely convex hulls aco(K α ), one will still be dealing with the same set of functionals f .…”