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I have some difficulty proving the following fundamental statement on the topic of topological groups; a topic that I have just started to study independently.

Let $G$ be a topological group. Consider an arbitrary element $g$ of $G$. It follows that a set $U$ is a neighborhood of $g$ if and only if $g^{-1}U$ is a neighborhood of the identity, $e$ of $G$.

I believe that the right approach will be along the line of, first defining specific continuous function $G\times G\to G$, exploiting that $G$ is a topological group. Then perhaps viewing the inverse of the function defined in order to deduce something on open sets. I know it is vogue, but this is the only thing that I can work with.

Julius Monk
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Consider the map from $G$ into itself defined by $h\mapsto g^{-1}h$. It is a homeomorphism which maps $g$ into $e$ (its inverse is $h\mapsto gh$), and therefore it maps $U$ into a neighborhood of $e$.

José Carlos Santos
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  • Just to deepen my understanding: If G was just a group, then the function defined is a group homomorphism, while if G was just a topological space, then this function would be a continuous map. In this combined structural situation, what makes them to behave like a homeomorphism? – Julius Monk Oct 19 '21 at 17:17
  • If $G$ was just a topological space, the map $h\mapsto g^{-1}h$ would make no sense. And it is a homeomorphism because it is a continuous bijection whose inverse is also continuous. That's why I mentioned that the inverse is the map $h\mapsto gh$. – José Carlos Santos Oct 19 '21 at 17:18
  • Perfect, it is clear to me now. One more question pops to my mind now: Is it possible to talk about the following structure, where G is a group, and U is a topological space, and study the action of G on U? And, speculation, define this action in a continuous manner? – Julius Monk Oct 19 '21 at 17:26
  • Yes. The standard definition is to say the the action is continuous if the map$$\begin{array}{ccc}G\times U&\longrightarrow&U\(g,u)&\mapsto&gu\end{array}$$is continuous. – José Carlos Santos Oct 19 '21 at 17:44
  • Thank you. I will return here when I will have enough reps to upvote. – Julius Monk Oct 19 '21 at 17:50