# Compact Topological Group Properties [closed]

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!

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## closed as off topic by Gerald Edgar, Qiaochu Yuan, Ryan Budney, Andreas Thom, Pete L. ClarkNov 14 '10 at 23:38

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Standard homework probably does not belong in MO. What else do you know about "compact Hausdorff" besides the definition? –  Gerald Edgar Nov 14 '10 at 22:44
The question has been closed. In level, its appropriateness is borderline (and thus it has received a correct answer). However, it has been phrased purely as a homework problem, with no other motivation, and such questions are discouraged on MO. (It would be a perfectly good question on math.stackexchange.com, for instance.) –  Pete L. Clark Nov 14 '10 at 23:40

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 :) )
Dear Mariano, I suppose there are easy counterexamples if $G$ is not compact, but I don't see any. Can you please help ? –  Georges Elencwajg Nov 15 '10 at 14:00