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


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


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.



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