nLab
weakly Lindelöf spaces with countably locally finite base are second countable

Statement

Recall that a topological space is weakly Lindelöf? if every open cover has a countable subcollection the union of which is dense.

Theorem

Every weakly Lindelöf? spaces with σ \sigma -locally finite base is second countable.

Proof

Let 𝒱\mathcal{V} be a countably locally finite base. For each xXx \in X, there is a neighborhood N xN_x meeting countably many members of 𝒱\mathcal{V}. If XX is weakly Lindelöf, there is a countable {N n} n\{N_n\}_n which covers a dense subset of XX. Then 𝒰={V𝒱N nV} \mathcal{U} = \{V\in \mathcal{V} \mid N_n \cap V \neq \emptyset\} is a countable basis for X.

propertiesimplications
second-countable: there is a countable base of the topology.A second-countable space has a σ \sigma -locally finite base: take the the collection of singeltons of all elements of countable cover of XX.
σ\sigma-locally finite base, i.e. XX has a countably locally finite base, e.g. a metrisable topological space by Nagata-Smirnov metrization theorem.second-countable spaces are separable: choose a point in each set of countable cover.
separable: there is a countable dense subset.second-countable spaces are Lindelöf
Lindelöf: every open cover has a countable sub-cover.weakly Lindelöf spaces with countably locally finite base are second countable
weakly Lindelöf: every open cover has a countable subcollection the union of which is dense.separable metacompact spaces are Lindelöf
countable choice: the natural numbers is a projective object in Set.separable spaces are weakly Lindelöf: given a countable dense subset and an open cover choose for each point of the subset an open from the cover.
metacompact: every open cover has a point-finite open refinement.Lindelöf spaces are trivially also weakly Lindelöf.

Created on April 2, 2019 at 12:17:27. See the history of this page for a list of all contributions to it.