On topological properties of Fréchet locally convex spaces with the weak topology

Authors:

• S.S. Gabriyelyan,
• Jerzy Kąkol,
• A. Kubzdela,
• M. Lopez-Pellicer

Abstract

We describe the topology of any cosmic space and any ℵ₀-space in terms of special bases defined by partially ordered sets. Using this description we show that a Baire cosmic group is metrizable. Next, we study those locally convex spaces (lcs) E which under the weak topology σ(E, E^') are ℵ₀-spaces. For a metrizable and complete lcs E not containing (an isomorphic copy of) ℓ₁ and satisfying the Heinrich density condition we prove that (E, σ(E, E^')) is an ℵ₀-space if and only if the strong dual of E is separable. In particular, if a Banach space E does not contain ℓ₁, then (E, σ(E, E^')) is an ℵ₀-space if and only if E^' is separable. The last part of the paper studies the question: Which spaces (E, σ(E, E^')) are ℵ₀-spaces? We extend, among the others, Michael's results by showing: If E is a metrizable lcs or a (DF)-space whose strong dual E^' is separable, then (E, σ(E, E^')) is an ℵ₀-space. Supplementing an old result of Corson we show that, for a Čech-complete Lindelöf space X the following are equivalent: (a) X is Polish, (b) C_c(X) is cosmic in the weak topology, (c) the weak^*-dual of C_c(X) is an ℵ₀-space.