every σ-compact set is Lindelöf

Theorem 1.

Every σ-compact (http://planetmath.org/SigmaCompact) set is Lindelöf (every open cover has a countableMathworldPlanetmath subcover).


Let X be a σ-compact. Let 𝒜 be an open cover of X . Since X is σ-compact, it is the union of countable many compact sets,


with Xi compact. Consider the cover 𝒜i={A𝒜:XiA} of the set Xi. This cover is well defined, it is not empty and covers Xi: for each xXi there is at least one of the open sets A𝒜 such that xA.

Since Xi is compact, the cover 𝒜i has a finite subcover. Then


and thus


That is, the set {Aij} is a countable subcover of 𝒜 that covers X. ∎

Title every σ-compact set is Lindelöf
Canonical name EverysigmacompactSetIsLindelof
Date of creation 2013-03-22 17:34:07
Last modified on 2013-03-22 17:34:07
Owner joen235 (18354)
Last modified by joen235 (18354)
Numerical id 14
Author joen235 (18354)
Entry type Theorem
Classification msc 54D45