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Let $\theta:G\times M\to M$ be a smooth left action of a Lie group $G$ on the manifold $M$. Suppose $G$ is compact and $M$ is Hausdorff. Let $K$ be a compact set in $M$. Is it true that $G_K:=\{g\in G:(g\cdot K)\cap K\neq \emptyset \}$ is closed?

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If $y_n \in G_K$ and $y_n \to y \in G$. Let $k_n, p_n \in K$ such that

$$y_n p_n = k_n$$

That can be found as $y_n \in G_K$. As $K$ is compact, by passing to subsequences we can assume that $k_n \to k$ and $p_n \to p$, where $k, p\in K$. Let $n\to \infty$, then $yp = k$ and $y\in G_K$.

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