# every second countable space is separable

###### Proof.

Let $X$ be a second countable space and let $\cal B$ be a countable base. For every non-empty set $B$ in $\cal B$, 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 EverySecondCountableSpaceIsSeparable 2013-03-22 12:22:10 2013-03-22 12:22:10 drini (3) drini (3) 5 drini (3) Proof msc 54-00 SecondCountable Separable