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

Abstract

A topological space X is strongly web-compact if X admits a family Aα : α ∈ N N of relatively countably compact sets covering X and such that Aα ⊂ A β for α ≤ β. The main result of this paper states the following: Theorem A Let X and Y be topological groups and f a homomorphism between X and Y with closed graph. If X is Fréchet-Urysohn and Baire and Y is strongly web-compact, then f is continuous. This extends a result of Valdivia. We provide an example showing that the property of being strongly web-compact is not productive. This applies to show that there are quasi-Suslin spaces X whose product X × X is not quasi-Suslin.

On C‐Suslin spaces

“…These results are stated below (the second in a slightly less general form), where all linear topological spaces are over the field of the real or complex numbers. Other extensions of Martineau's theorem for topological groups can be found in , . Theorem Every linear mapping with closed graph from a Baire linear topological space into a K ‐Suslin linear topological space is continuous. Theorem Every linear mapping with closed graph from a metrizable Baire linear topological space into a C ‐Suslin linear topological space is continuous. Theorem Every linear mapping with closed graph from a locally convex hull of metrizable Baire locally convex spaces into a quasi‐Suslin locally convex space is continuous. Theorem Every linear mapping with closed graph from a Baire linear topological space into a linear topological space which admits a relatively countably compact resolution is continuous.…”

“…such that whenever for all . Spaces that admit a relatively countably compact resolution have been called strongly web‐compact in (see also [, Chapter 5] for details). A topological space X is called C ‐ Suslin if there is a map T from a separable metric space P into such that covers X and if is a Cauchy sequence in P then is a relatively countably compact set in X .…”

“…Valdivia showed [, I.4.3 (23, 24)] that if E is a Fréchet space, its weak* bidual is always quasi‐Suslin, but is K ‐analytic if and only if is barrelled. It is shown in [, Example 13] that if (the bounding cardinal) then the weak* dual F of the space consisting of all real‐valued continuous functions defined on the ordinal interval [0, ω 1 ) equipped with the compact‐open topology is quasi‐Suslin, hence C ‐Suslin; but since F is not Lindelöf (see for instance ), then F is not K ‐analytic. Other examples of C ‐Suslin spaces which are not K ‐analytic can be found in .…”

“…Let us provide a proof of this fact. According to the proof of [, Proposition 1.1], if X is covered by an ordered family of relatively countably compact sets, given and with for , then each sequence in X such that for every has a cluster point in X . Hence, defining by it happens that , and if in and we claim that has a cluster point in X .…”

“…All completely regular spaces X with a countable network modulo a cover of X by compact sets are Lindelöf Σ‐spaces (see [, Proposition IV.9.1]). Example There exists a C ‐Suslin space G such that is not a C ‐Suslin space. Proof In [, Example 2.1] it is shown that if X is a discrete space of cardinality there exists a countably compact topological space G such that X is homeomorphic to a closed subspace of the product . Since G is countably compact, G is quasi‐Suslin and hence C ‐Suslin.…”