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While reading tom Dieck's Algebraic Topology, I've come across to the fact which mention that a homomorphism $\phi\colon G\to H$ of topological groups is continuous if it is continuous at $e_G$. I couldn't come up with a proof for this, so I started searching. It seems we can take any $g \in G$ instead of $e_G$ and that the fact has something to do with $\lambda_g\colon x \mapsto gx$ being homeomorphisms. But I still don't see how to prove this. I'm a little afraid to ask this since it fells like I'm missing something trivial.

Jxt921
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  • Does the second answer here: https://math.stackexchange.com/questions/1318867/continuous-homorphisms-between-topological-groups answer your question? – Louis Hainaut Mar 04 '20 at 12:41
  • @LouisHainaut Yeah, it does in fact answer my question, thank you! – Jxt921 Mar 04 '20 at 12:56

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Yes, this follows directly from the fact that all $\lambda_g$ are homeomorphisms of $G$ and have the property that $\lambda_g(e)=g$ as well.

Suppose $h: G \to G$ is a group homeomorphism which is continuous at $e$.

Then take $x \in G$ and $y=h(x)$. To see that $h$ is continuous at $x$ note that

$$\lambda_y \circ h \circ \lambda_{x^{-1}} = h$$ because $h$ is a homeomorphism ($yh(x^{-1}g) = h(x)h(x^{-1})h(g) = h(g)$ for all $g \in G$)

and analysing pointwise continuity: $\lambda_{x^{-1}}$ is continuous at $x$ (everywhere actually), maps $x$ to $e$, $h$ is continuous at $e$ (and maps $e$ to $e$) and $\lambda_y$ is continuous at $e$ (everywhere even) so in total $h$ is continuous at $x$, as required.

A solution using nets can be found similarly.

Henno Brandsma
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