If a topological space $X$ is locally compact and I have that $x in C$ for $C$ compact, is there a $U$ of $x$...
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The definition of a locally compact topological space $X$, according to my notes, is that for every $x in X$, there exists compact $C subset X$ such that $x in C$ with a neighbourhood $U$ of $x$ with $x in U subset C$.
My question is a slight variation of this. If a topological space $X$ is locally compact and I have that $x in C$ for $C$ compact, does there exist a neighbourhood $U$ of $x$ such that $x in U subset C$. The difference is now I am starting with a compact set.
general-topology
asked Nov 15 at 3:33
IntegrateThis
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The definition of a locally compact topological space $X$, according to my notes, is that for every $x in X$, there exists compact $C subset X$ such that $x in C$ with a neighbourhood $U$ of $x$ with $x in U subset C$.
My question is a slight variation of this. If a topological space $X$ is locally compact and I have that $x in C$ for $C$ compact, does there exist a neighbourhood $U$ of $x$ such that $x in U subset C$. The difference is now I am starting with a compact set.
general-topology
asked Nov 15 at 3:33
IntegrateThis
1,7021717
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
The definition of a locally compact topological space $X$, according to my notes, is that for every $x in X$, there exists compact $C subset X$ such that $x in C$ with a neighbourhood $U$ of $x$ with $x in U subset C$.
My question is a slight variation of this. If a topological space $X$ is locally compact and I have that $x in C$ for $C$ compact, does there exist a neighbourhood $U$ of $x$ such that $x in U subset C$. The difference is now I am starting with a compact set.
general-topology
asked Nov 15 at 3:33
IntegrateThis
1,7021717
The definition of a locally compact topological space $X$, according to my notes, is that for every $x in X$, there exists compact $C subset X$ such that $x in C$ with a neighbourhood $U$ of $x$ with $x in U subset C$.
My question is a slight variation of this. If a topological space $X$ is locally compact and I have that $x in C$ for $C$ compact, does there exist a neighbourhood $U$ of $x$ such that $x in U subset C$. The difference is now I am starting with a compact set.
general-topology
general-topology
asked Nov 15 at 3:33
IntegrateThis
1,7021717
asked Nov 15 at 3:33
IntegrateThis
1,7021717
asked Nov 15 at 3:33
IntegrateThis
1,7021717
asked Nov 15 at 3:33
IntegrateThis
1,7021717
asked Nov 15 at 3:33
IntegrateThis
1,7021717
1,7021717
add a comment |
add a comment |
2 Answers
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Take $X = mathbb{R}$, $x in X$ and ${x} = C$.
answered Nov 15 at 3:47
Bias of Priene
28612
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But $C$ might have no interior.
For example, let $X=mathbb{R}^2$, let $x$ be any point of $X$, and let $C$ be a line segment through $x$.
Even if $C$ has interior, the point $x$ might be on the boundary of $C$.
For example, using $X=mathbb{R}^2$ again, let $x$ be any point of $X$, and let $C$ be a closed disk whose circular boundary passes through $x$.
answered Nov 15 at 3:44
quasi
35.9k22562
add a comment |
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
2
down vote
accepted
Take $X = mathbb{R}$, $x in X$ and ${x} = C$.
answered Nov 15 at 3:47
Bias of Priene
28612
add a comment |
up vote
2
down vote
accepted
Take $X = mathbb{R}$, $x in X$ and ${x} = C$.
answered Nov 15 at 3:47
Bias of Priene
28612
add a comment |
up vote
2
down vote
accepted
up vote
2
down vote
accepted
Take $X = mathbb{R}$, $x in X$ and ${x} = C$.
answered Nov 15 at 3:47
Bias of Priene
28612
Take $X = mathbb{R}$, $x in X$ and ${x} = C$.
answered Nov 15 at 3:47
Bias of Priene
28612
answered Nov 15 at 3:47
Bias of Priene
28612
answered Nov 15 at 3:47
Bias of Priene
28612
answered Nov 15 at 3:47
Bias of Priene
28612
28612
add a comment |
add a comment |
up vote
1
down vote
But $C$ might have no interior.
For example, let $X=mathbb{R}^2$, let $x$ be any point of $X$, and let $C$ be a line segment through $x$.
Even if $C$ has interior, the point $x$ might be on the boundary of $C$.
For example, using $X=mathbb{R}^2$ again, let $x$ be any point of $X$, and let $C$ be a closed disk whose circular boundary passes through $x$.
answered Nov 15 at 3:44
quasi
35.9k22562
add a comment |
up vote
1
down vote
But $C$ might have no interior.
For example, let $X=mathbb{R}^2$, let $x$ be any point of $X$, and let $C$ be a line segment through $x$.
Even if $C$ has interior, the point $x$ might be on the boundary of $C$.
For example, using $X=mathbb{R}^2$ again, let $x$ be any point of $X$, and let $C$ be a closed disk whose circular boundary passes through $x$.
answered Nov 15 at 3:44
quasi
35.9k22562
add a comment |
up vote
1
down vote
up vote
1
down vote
But $C$ might have no interior.
For example, let $X=mathbb{R}^2$, let $x$ be any point of $X$, and let $C$ be a line segment through $x$.
Even if $C$ has interior, the point $x$ might be on the boundary of $C$.
For example, using $X=mathbb{R}^2$ again, let $x$ be any point of $X$, and let $C$ be a closed disk whose circular boundary passes through $x$.
answered Nov 15 at 3:44
quasi
35.9k22562
But $C$ might have no interior.
For example, let $X=mathbb{R}^2$, let $x$ be any point of $X$, and let $C$ be a line segment through $x$.
Even if $C$ has interior, the point $x$ might be on the boundary of $C$.
For example, using $X=mathbb{R}^2$ again, let $x$ be any point of $X$, and let $C$ be a closed disk whose circular boundary passes through $x$.
answered Nov 15 at 3:44
quasi
35.9k22562
edited Nov 15 at 3:49
answered Nov 15 at 3:44
quasi
35.9k22562
answered Nov 15 at 3:44
quasi
35.9k22562
answered Nov 15 at 3:44
quasi
35.9k22562
35.9k22562
add a comment |
add a comment |
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