Every element $g$ of $G$ has a symmetric neighborhood $V$ of $e$ such that $VgV^{-1}subset U$
Let $G$ be a topological group and $gin G$ , $U$ is a neighborhood of $g$ . Prove that there exists a symmetric neighborhood $V$ of $e$ such that $VgV^{-1}subset U$.
If $g=e$, l have proved it. But if $gneq e$ , l have no idea. So how to prove ?
topological-groups
asked Nov 27 '18 at 9:44
water graph
232
add a comment |
Let $G$ be a topological group and $gin G$ , $U$ is a neighborhood of $g$ . Prove that there exists a symmetric neighborhood $V$ of $e$ such that $VgV^{-1}subset U$.
If $g=e$, l have proved it. But if $gneq e$ , l have no idea. So how to prove ?
topological-groups
asked Nov 27 '18 at 9:44
water graph
232
Welcome to MSE. What is $U$?
– José Carlos Santos
Nov 27 '18 at 9:48
$U$ is any neighborhood of $g$.
– water graph
Nov 27 '18 at 9:51
add a comment |
Let $G$ be a topological group and $gin G$ , $U$ is a neighborhood of $g$ . Prove that there exists a symmetric neighborhood $V$ of $e$ such that $VgV^{-1}subset U$.
If $g=e$, l have proved it. But if $gneq e$ , l have no idea. So how to prove ?
topological-groups
asked Nov 27 '18 at 9:44
water graph
232
Let $G$ be a topological group and $gin G$ , $U$ is a neighborhood of $g$ . Prove that there exists a symmetric neighborhood $V$ of $e$ such that $VgV^{-1}subset U$.
If $g=e$, l have proved it. But if $gneq e$ , l have no idea. So how to prove ?
topological-groups
topological-groups
asked Nov 27 '18 at 9:44
water graph
232
asked Nov 27 '18 at 9:44
water graph
232
asked Nov 27 '18 at 9:44
water graph
232
asked Nov 27 '18 at 9:44
water graph
232
asked Nov 27 '18 at 9:44
water graph
232
232
Welcome to MSE. What is $U$?
– José Carlos Santos
Nov 27 '18 at 9:48
$U$ is any neighborhood of $g$.
– water graph
Nov 27 '18 at 9:51
add a comment |
Welcome to MSE. What is $U$?
– José Carlos Santos
Nov 27 '18 at 9:48
$U$ is any neighborhood of $g$.
– water graph
Nov 27 '18 at 9:51
Welcome to MSE. What is $U$?
– José Carlos Santos
Nov 27 '18 at 9:48
Welcome to MSE. What is $U$?
– José Carlos Santos
Nov 27 '18 at 9:48
$U$ is any neighborhood of $g$.
– water graph
Nov 27 '18 at 9:51
$U$ is any neighborhood of $g$.
– water graph
Nov 27 '18 at 9:51
add a comment |
1 Answer
1
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Let $f: G to G$ be $f(x) = xgx^{-1}$. Then $f(e) = g$. So by continuity, we can find some open neighborhood $V$ of $e$ for which $f(V) subset U$, i.e. for which $VgV^{-1} subset U$. To make $V$ symmetric, we can just replace $V$ with $Vcap V^{-1}$, which is a subset of $V$ (so that $VgV^{-1} subset U$ still holds).
answered Nov 27 '18 at 9:51
mathworker21
8,6371928
add a comment |
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1 Answer
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1 Answer
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Let $f: G to G$ be $f(x) = xgx^{-1}$. Then $f(e) = g$. So by continuity, we can find some open neighborhood $V$ of $e$ for which $f(V) subset U$, i.e. for which $VgV^{-1} subset U$. To make $V$ symmetric, we can just replace $V$ with $Vcap V^{-1}$, which is a subset of $V$ (so that $VgV^{-1} subset U$ still holds).
answered Nov 27 '18 at 9:51
mathworker21
8,6371928
add a comment |
Let $f: G to G$ be $f(x) = xgx^{-1}$. Then $f(e) = g$. So by continuity, we can find some open neighborhood $V$ of $e$ for which $f(V) subset U$, i.e. for which $VgV^{-1} subset U$. To make $V$ symmetric, we can just replace $V$ with $Vcap V^{-1}$, which is a subset of $V$ (so that $VgV^{-1} subset U$ still holds).
answered Nov 27 '18 at 9:51
mathworker21
8,6371928
add a comment |
Let $f: G to G$ be $f(x) = xgx^{-1}$. Then $f(e) = g$. So by continuity, we can find some open neighborhood $V$ of $e$ for which $f(V) subset U$, i.e. for which $VgV^{-1} subset U$. To make $V$ symmetric, we can just replace $V$ with $Vcap V^{-1}$, which is a subset of $V$ (so that $VgV^{-1} subset U$ still holds).
answered Nov 27 '18 at 9:51
mathworker21
8,6371928
Let $f: G to G$ be $f(x) = xgx^{-1}$. Then $f(e) = g$. So by continuity, we can find some open neighborhood $V$ of $e$ for which $f(V) subset U$, i.e. for which $VgV^{-1} subset U$. To make $V$ symmetric, we can just replace $V$ with $Vcap V^{-1}$, which is a subset of $V$ (so that $VgV^{-1} subset U$ still holds).
answered Nov 27 '18 at 9:51
mathworker21
8,6371928
answered Nov 27 '18 at 9:51
mathworker21
8,6371928
answered Nov 27 '18 at 9:51
mathworker21
8,6371928
answered Nov 27 '18 at 9:51
mathworker21
8,6371928
8,6371928
add a comment |
add a comment |
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Welcome to MSE. What is $U$?
– José Carlos Santos
Nov 27 '18 at 9:48
$U$ is any neighborhood of $g$.
– water graph
Nov 27 '18 at 9:51