second-countable spaces are Lindelöf



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Using countable choice, then: Every second-countable topological space XX is Lindelöf, i.e. any open cover admits a countable subcover.


Let {U n}\{U_n\} be a countable base of the topology. Given any open cover {V λ}\{V_\lambda\} of XX, we can form the index set II\subset \mathbb{N} of those nn that are contained in some V λV_\lambda. By assumption iIU i= λV λ=X\bigcup_{i\in I} U_{i} = \bigcup_\lambda V_\lambda = X. The axiom of countable choice provides now a section of iI{λU iV λ}I\bigsqcup_{i\in I} \{\lambda \mid U_i \subset V_\lambda\}\to I.

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.

Last revised on April 2, 2019 at 12:23:03. See the history of this page for a list of all contributions to it.