I feel I want to understand it better. I know that for every cover there's a finite subcover but what can you say about it's group properties?

I'm stuck on this homework problem where we were asked: Let G compact, Hausdorff which has the structure of a group. And multiplication $m: G \times G \rightarrow G$ is continuous. Show G is a topological group.

All I need to do is to show the inverse map, $ inv: G\to G $ is continuous. So I have to somehow use multiplication is continuous and the fact it is compact to show inv is continuous.

I tried to reason $inv(x) = x^{-1}$ as $L_{x^{-2}}(x) = x^{-2}*x = x^{-1}$ and try to say something with continuity since $L_{x^{-2}}$ is continuous, guaranteed by multiplication is continuous. But $inv$ becomes too independent on a particular x it operates on... So I don't know.

I think if I just know compact topological groups better I'd be in okay shape.

Thanks!