every second countable space is separable

Theorem 1.

Let X be a second countable space and let be a countable base. For every non-empty set B in , choose a point xBB. The set A of all such points xB is clearly countableMathworldPlanetmath and it’s also dense since any open set intersects it and thus the whole space is the closureMathworldPlanetmathPlanetmath of A. That is, A is a countably dense subset of X. Therefore, X is separable. ∎


  • 1 J.L. Kelley, General Topology, D. van Nostrand Company, Inc., 1955.
Title every second countable space is separable
Canonical name EverySecondCountableSpaceIsSeparable
Date of creation 2013-03-22 12:22:10
Last modified on 2013-03-22 12:22:10
Owner drini (3)
Last modified by drini (3)
Numerical id 5
Author drini (3)
Entry type Proof
Classification msc 54-00
Related topic SecondCountable
Related topic Separable