Topological Group $G$ totally disconnected $\Rightarrow$ $G$ hausdorff?

Question

On Wikipedia, I read that a topological group is necessarily Hausdorff if it is totally disconnected. Is that true? I read it on this page:

http://en.wikipedia.org/wiki/Totally_disconnected_group

If not, does anyone has an example for a totally disconnected group, which is not Hausdorff?

Thanks for help.

Answer 1

It's true. If $G$ is a totally disconnected topological group, then the connected component of the identity $e$ is $\{e\}$ (because components of a totally disconnected space are just points). Since connected components of any topological space are closed, $\{e\}$ is closed. This implies that $G$ is Hausdorff: the the diagonal $\Delta\subseteq G\times G$ is the inverse image of $\{e\}$ under the continuous map $(g,h)\mapsto gh^{-1}$, and is therefore closed.

Answer 2

A space $X$ is Hausdorff if and only if the diagonal $\Delta \subseteq X \times X$ is closed. In the case of a topological group, the diagonal can be described as the preimage of $\{1\}$ under the map $X\times X \to X : (x, y) \mapsto xy^{-1}$, and a point is closed in a disconnected space.