## Abstract

We describe the topology of any cosmic space and any ℵ_{0}-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 ℵ_{0}-spaces. For a metrizable and complete lcs E not containing (an isomorphic copy of) ℓ_{1} and satisfying the Heinrich density condition we prove that (E, σ(E, E^{'})) is an ℵ_{0}-space if and only if the strong dual of E is separable. In particular, if a Banach space E does not contain ℓ_{1}, then (E, σ(E, E^{'})) is an ℵ_{0}-space if and only if E^{'} is separable. The last part of the paper studies the question: Which spaces (E, σ(E, E^{'})) are ℵ_{0}-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 ℵ_{0}-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 ℵ_{0}-space.

Original language | English |
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Pages (from-to) | 123-137 |

Number of pages | 15 |

Journal | Topology and its Applications |

Volume | 192 |

DOIs | |

State | Published - 1 Sep 2015 |

## Keywords

- Banach space
- K-network
- Locally convex Fréchet space
- Weak topology
- ℵ-space