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

We describe the topology of any cosmic space and any ℵ0ℵ0-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′)σ(E,E′) are ℵ0ℵ0-spaces. For a metrizable and complete lcs E   not containing (an isomorphic copy of) ℓ1ℓ1 and satisfying the Heinrich density condition we prove that (E,σ(E,E′))(E,σ(E,E′)) is an ℵ0ℵ0-space if and only if the strong dual of E is separable. In particular, if a Banach space E   does not contain ℓ1ℓ1, then (E,σ(E,E′))(E,σ(E,E′)) is an ℵ0ℵ0-space if and only if E′E′ is separable. The last part of the paper studies the question: Which spaces (E,σ(E,E′))(E,σ(E,E′)) are ℵ0ℵ0-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′E′ is separable, then (E,σ(E,E′))(E,σ(E,E′)) is an ℵ0ℵ0-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) Cc(X)Cc(X) is cosmic in the weak topology, (c) the weak⁎weak⁎-dual of Cc(X)Cc(X) is an ℵ0ℵ0-space.