Some questions about topological groups

[R136a1]

So, I have a topological group $G$. This means that the functions

$$m:G\times G\rightarrow G:(x,y)\rightarrow xy$$

and

$$i:G\rightarrow G:x\rightarrow x^{-1}$$

are continuous.

I have a couple of questions that seem mysterious to me.

Let's start with this: I've seen a statement somewhere that says that if $H$ is a subgroup of $G$ such that $H$ is open, then $H$ is actually also closed. How do I prove such a thing? I know that $\overline{H}$ is a subgroup, do I prove that $H=\overline{H}$?

 

[jgens]

Use the openness of H combined with the fact that you can write G as a disjoint union of the cosets of H.

 

[R136a1]

Ah yes, I got it now. That's really a nice technique! Thanks a lot.