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second-countable spaces are Lindelöf

Context

Topology

topology (point-set topology, point-free topology)

see also differential topology, algebraic topology, functional analysis and topological homotopy theory

Introduction

Basic concepts

Universal constructions

Extra stuff, structure, properties

Examples

Basic statements

Theorems

Analysis Theorems

topological homotopy theory

Statement

Theorem

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

Proof

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.

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.

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