nLab
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.

Axioms: axiom of choice (AC), countable choice (CC).

Properties

Implications

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