Group topology induced by group action Let $G$ be a group, $X$ a topological space and $G\times X \longrightarrow X$ a group action so that $\begin{align*}
X\longrightarrow X \\
x \mapsto g\cdot x
\end{align*}$ is a homeomorphism $\forall g \in G$. Is there a coarsest topology on $G$ so that:

*

*$G$ is a topological group

*The group action is continuous

?
I have showed that the topology induced by the maps $\begin{align*} G \longrightarrow X \\
g\mapsto g\cdot x\end{align*}$ and
$\begin{align*} G \longrightarrow X \\
g\mapsto g^{-1}\cdot x\end{align*}$ $\forall x\in X$ makes inversion, left action and right action continuous on $G$.
 A: If $X$ is hausdorff and has compact and connected neighbourhoods around each point, the answer is the topology induced by the homomorphism $G\longrightarrow \text{Homeo}(X)$ defined by the group action, where $\text{Homeo}(X)$ is the homeomorphism group equipped with the compact-open-topology. To prove it:
Let $\tau$ be a topology on $G$ so that $G$ is a topological group and the group action $G\times X\longrightarrow X$ is continuous. Since $X$ is hausdorff and locally compact, the exponential object $X^X$ exists in the category $\text{Top}$. By the universal property of exponential objects $\exists!$ continuous $G \overset{\phi }{\longrightarrow} X^X$ so that

commutes. $X^X$ can be constructed as $X^X=C(X,X)$ equipped with the compact-open-topology. $\text{Homeo}(X)\subseteq X^X$. $\text{Im }\phi\subseteq\text{Homeo}(X)$. $G\overset{\phi}{\longrightarrow}\text{Homeo}(X)$ is precisely the homomorphism defined by the group action. There's a theorem stating that if $X$ is hausdorff and has compact and connected neighbourhoods around each point, $\text{Homeo}(X)$ equipped with the compact-open-topology is a topological group https://www.jstor.org/stable/pdf/30037630.pdf?refreqid=excelsior%3A0a25ff84e89d996eeaf57120097c8447. Define $\tau_i$ to be the topology on $G$ induced by $G\overset{\phi}{\longrightarrow}\text{Homeo}(X)$. Then $(G,\tau_i)$ is a topological group from the universl property of induced topology and since $\phi$ is a homomorphism. Since $(G,\tau)\overset{\phi}{\longrightarrow} \text{Homeo}(X)$ is continuous, $\tau_i\subseteq \tau$. That means $\tau_i$ is the coarsest topology so the action is continuous and $G$ is a topological group.
