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 $\mathcal{B}$ be a countable base. For every non-empty set $B$ in $\mathcal{B}$, 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. ∎
References
- 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 |