every second countable space is separable

Theorem 1.

[1] Every second countable space is separable.

Proof.

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

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