proof of Lindelöf theorem

Let X be a second countable topological spaceMathworldPlanetmath, AX any subset and 𝒰 an open cover of A. Let be a countable basis for X; then ={BA:B} is a countable basis of the subspace topology on A. Then for each aA there is some Ua𝒰 with aUa, and so there is Ba such that aBaUa.

Then {Ba:aA} is a countable open cover of A. For each Ba, choose UBa𝒰 such that BaUBa. Then {UBa:aA} is a countable subcover of A from 𝒰.

Title proof of Lindelöf theorem
Canonical name ProofOfLindelofTheorem
Date of creation 2013-03-22 12:56:31
Last modified on 2013-03-22 12:56:31
Owner Evandar (27)
Last modified by Evandar (27)
Numerical id 5
Author Evandar (27)
Entry type Proof
Classification msc 54D99