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limit points of a subset of a compact set

Verification of the proof of the theorem: “E is infinite subset of a compact set K, prove: E has a limit point in K”Let X be a metric space in which every infinite subset has a limit point. Prove that X is compact.Thinking Process: a set is closed if it contains all of its limit points (--> this direction)Limit point of an infinite subset of a compact setCompact subset of $mathbb{R}^1$ with countable limit pointsProve a metric space in which every infinite subset has a limit point is compact.(Question about one particular proof)Limit Points for a SetRudin's theorem on compact sets and limit pointsProving that a finite point set is closed by using limit pointsReal Analysis - Limit points and Open set.

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so I'm confused. I know that any infinite subset of a compact set $X$ should have a limit point in $X$. I wonder why it is not true that all the limit points of any subset $E$ of a compact set $X$ belong to $X$? If any limit point $p$ of $E$ does not belong to $X$, it means that it belongs to the complement of $X$ which is an open set. Then there is a neighborhood around $p$ which has no intersection with $X$ therefore not with $E$ too, contradicting the definition of a limit point. Where am I mistaken? thx in advance

real-analysis

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asked Mar 14 at 12:27

shota kobakhidzeshota kobakhidze

115

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  You are not mistaken, in fact for any closed set $X$ the limit points of a subset $Esubset X$ are contained in $X$.
  $endgroup$
  – Floris Claassens
  Mar 14 at 12:37
  

  
  
  
  


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  Is the space you're considering Hausdorff or a general topological space?
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  – Chrystomath
  Mar 14 at 12:38
  

  
  
  
  


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  @Chrystomath: It's a general topological space. Have no experience with Hausdorff space to be honest.
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  – shota kobakhidze
  Mar 14 at 12:43
  

  
  
  
  


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  @FlorisClaassens: I was just confused why Rudin had it as a theorem, I thought the extension of the theorem to any subset(not only infinite ones) was false.
  $endgroup$
  – shota kobakhidze
  Mar 14 at 12:47
  
  
  

  
  
  
  


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  @shota kobakhidze, I did some quick extra research, and this has a lot to do with the definition of a limit point. Some point $x$ is a limit point of $E$ if every neighbourhood of $x$ contains a point of $E$ other then $x$ itself. By this definition, clearly if $E$ is finite it has no limit points.
  $endgroup$
  – Floris Claassens
  Mar 14 at 13:01
  

  
  
  
  

 | 
show 6 more comments

0

$begingroup$

so I'm confused. I know that any infinite subset of a compact set $X$ should have a limit point in $X$. I wonder why it is not true that all the limit points of any subset $E$ of a compact set $X$ belong to $X$? If any limit point $p$ of $E$ does not belong to $X$, it means that it belongs to the complement of $X$ which is an open set. Then there is a neighborhood around $p$ which has no intersection with $X$ therefore not with $E$ too, contradicting the definition of a limit point. Where am I mistaken? thx in advance

real-analysis

share|cite|improve this question

asked Mar 14 at 12:27

shota kobakhidzeshota kobakhidze

115

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  You are not mistaken, in fact for any closed set $X$ the limit points of a subset $Esubset X$ are contained in $X$.
  $endgroup$
  – Floris Claassens
  Mar 14 at 12:37
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Is the space you're considering Hausdorff or a general topological space?
  $endgroup$
  – Chrystomath
  Mar 14 at 12:38
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @Chrystomath: It's a general topological space. Have no experience with Hausdorff space to be honest.
  $endgroup$
  – shota kobakhidze
  Mar 14 at 12:43
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @FlorisClaassens: I was just confused why Rudin had it as a theorem, I thought the extension of the theorem to any subset(not only infinite ones) was false.
  $endgroup$
  – shota kobakhidze
  Mar 14 at 12:47
  
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @shota kobakhidze, I did some quick extra research, and this has a lot to do with the definition of a limit point. Some point $x$ is a limit point of $E$ if every neighbourhood of $x$ contains a point of $E$ other then $x$ itself. By this definition, clearly if $E$ is finite it has no limit points.
  $endgroup$
  – Floris Claassens
  Mar 14 at 13:01
  

  
  
  
  

 | 
show 6 more comments

$begingroup$

so I'm confused. I know that any infinite subset of a compact set $X$ should have a limit point in $X$. I wonder why it is not true that all the limit points of any subset $E$ of a compact set $X$ belong to $X$? If any limit point $p$ of $E$ does not belong to $X$, it means that it belongs to the complement of $X$ which is an open set. Then there is a neighborhood around $p$ which has no intersection with $X$ therefore not with $E$ too, contradicting the definition of a limit point. Where am I mistaken? thx in advance

real-analysis

share|cite|improve this question

asked Mar 14 at 12:27

shota kobakhidzeshota kobakhidze

115

$endgroup$

share|cite|improve this question

asked Mar 14 at 12:27

shota kobakhidzeshota kobakhidze

115

share|cite|improve this question

asked Mar 14 at 12:27

shota kobakhidzeshota kobakhidze

115

asked Mar 14 at 12:27

shota kobakhidzeshota kobakhidze

115

asked Mar 14 at 12:27

shota kobakhidzeshota kobakhidze

115