Mixing
invariant neighbourhood under a continuous group action
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.
general-topology compactness group-actions
asked Nov 20 '18 at 2:59
Vitolo
404
add a comment |
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.
general-topology compactness group-actions
asked Nov 20 '18 at 2:59
Vitolo
404
add a comment |
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.
general-topology compactness group-actions
asked Nov 20 '18 at 2:59
Vitolo
404
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.
general-topology compactness group-actions
general-topology compactness group-actions
asked Nov 20 '18 at 2:59
Vitolo
404
asked Nov 20 '18 at 2:59
Vitolo
404
asked Nov 20 '18 at 2:59
Vitolo
404
asked Nov 20 '18 at 2:59
Vitolo
404
asked Nov 20 '18 at 2:59
Vitolo
404
404
add a comment |
add a comment |
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