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I am learning about the compact-open topology and have a small proposition I am struggling to prove. Let $G$ be a topological group, $X$ a compact, Hausdorff space, and $H(X)$, the homeomorphisms of $X$, have the compact open topology. I want to show that an action of $G$ on $X$, call it $\gamma_1:G\times X\rightarrow X$ is continuous iff its associated homomorphism $\gamma_2:G\rightarrow H(X)$, where $g\mapsto \phi_g$ (left-translation by $g$), is continuous.

I can do the $\Leftarrow$ direction: assuming $\gamma_2$ is continuous, the map $(g,x)\mapsto (\phi_g,x)$ is continuous. Since the evaluation map $(\phi_g,x)\mapsto \phi_g(x)$ is continuous, and $\gamma_1$ is the composition of these two, $\gamma_2$ is continuous.

The other direction is where I'm not sure how to proceed. We can take $\phi_g\in H(X)$ and a subbasis set $S(C,U)=\{f:f(C)\subseteq U\}$ for $C$ compact, $U$ open, such that $\phi_g\in S(C,U)$. So $\gamma_2^{-1}(S(C,U))=\{h\in G:h\cdot C\subseteq U\}$. This is all true, but I am not sure it is helpful. I am not sure what the right approach is; in particular I don't see how/when to leverage the fact that $\gamma_1$ is continuous.

Thanks for any hints, direction, insight, etc.

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I think, it is quite important to carefully write down what one is supposed to do in a situation like this one, as it is really easy to get confused.

You are given a subbasis element $S(C,U)$ with $C\subset X$ compact, and $U\subset X$ open. To prove continuity of $\gamma_2$ it suffices to show that for any $g\in G$ satisfying $g(C)\subset U$ there exists a neighbourhood $V\subset G$ of $g$ such that $$ h\in V \implies h(C) \subset U$$ Luckily we are only given one information about the action of $G$ on $X$, so we immediately now where to start: Fix $g\in G$ as above. We know that for any $x\in C$ we have $g(x) \in U$. Also fix $x$ for the moment.

By assumption on the continuity of $\gamma_1$ there are neighbourhoods $W_x \subset X$ of $x$ and $V_x\subset G$ of $g$ such that $$(h,y)\in V_x\times W_x \implies h(y) = \gamma_1(h,y)\in U$$ Now finitely many of these $W_x$ cover $C$, say $C\subset W_{x_1}\cup\dots\cup W_{x_n}$.

Let $V = V_{x_1}\cap \dots \cap V_{x_n}$. Then $V$ is an open neighbourhood of $g$ in $G$ and I'll leave it to you to verify that $\gamma_2(V)\subset S(C,U)$.

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