Compact Topological Group Properties [closed]

Question

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!

Answer 1

Consider the function $\beta:(x,y)\in G\times G\mapsto (x,xy)\in G\times G$. It is continuous and bijective, as you can easily check. It follows from your hypotheses too that $\beta$ is an homeomorphism. If now we define the functions $\lambda:x\in G\mapsto (x,1)\in G\times G$ and $\pi:(x,y)\in G\times G\mapsto y\in G$, which are obviously continuous, then the composition $\iota=\pi\circ\beta^{-1}\circ\lambda:G\to G$ is also continuous.

Magically, $\iota$ is the inversion map.

(I learnt this argument from reading the extraordinary proof by Peter Schauenburg that a bialgebra $H$ over a field $k$ for which there exists an $H$-Galois extension of $k$ is in fact a Hopf algebra. In a way, this is the non-non-commutative version :) )