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I found this statement and I am really struggling trying to come up with a proof of it. The situation is the following:

Let $G$ be a compact topological group acting continuously on a compact Hausdorff space $X$. Then for any $x \in X$ there is a $G$-invariant neighbourhood $U$ of $X$ such that for any $y \in U$, $gG_yg^{-1} \subseteq G_x$ for some $g \in G$. (Here $G_x$ denotes the stabilizer subgroup of $X$)

I think that the slice theorem might be useful here (There is a $G$-invariant neighbourhood of $X$ of the form $(G \times A)/G_x$ for some $G_x$-invariant subset $A$ of $X$) but I do not see how to proceed using this.

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