# Hausdorff space

we know that each infinite subspace of a KC space contain an infinite discrete subspace

The Hausdorff spaces imply KC spaces.But:

Is it right to say that every Hausdorff space has an infinite discrete subspace?

• Any finite discrete space is Hausdorff, but obviously can't contain an infinite discrete subspace. – user38355 Aug 6 '13 at 6:35

It is true that every infinite Hausdorff space $X$ has an infinite discrete (in itself) subspace. The extra condition that $X$ is infinite itself is clearly necessary... And if you know the fact for KC spaces, it certainly follows for Hausdorff spaces.