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.










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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.










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      up vote
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      down vote

      favorite









      up vote
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      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.










      share|cite|improve this question













      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.







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      asked Nov 15 at 3:33









      IntegrateThis

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          Take $X = mathbb{R}$, $x in X$ and ${x} = C$.






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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 $x$.






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              2 Answers
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              2 Answers
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              active

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              active

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              up vote
              2
              down vote



              accepted










              Take $X = mathbb{R}$, $x in X$ and ${x} = C$.






              share|cite|improve this answer

























                up vote
                2
                down vote



                accepted










                Take $X = mathbb{R}$, $x in X$ and ${x} = C$.






                share|cite|improve this answer























                  up vote
                  2
                  down vote



                  accepted







                  up vote
                  2
                  down vote



                  accepted






                  Take $X = mathbb{R}$, $x in X$ and ${x} = C$.






                  share|cite|improve this answer












                  Take $X = mathbb{R}$, $x in X$ and ${x} = C$.







                  share|cite|improve this answer












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                  answered Nov 15 at 3:47









                  Bias of Priene

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                      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 $x$.






                      share|cite|improve this answer



























                        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 $x$.






                        share|cite|improve this answer

























                          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 $x$.






                          share|cite|improve this answer














                          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 $x$.







                          share|cite|improve this answer














                          share|cite|improve this answer



                          share|cite|improve this answer








                          edited Nov 15 at 3:49

























                          answered Nov 15 at 3:44









                          quasi

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