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What are the limit points of $X times mathbbN$?

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0

$begingroup$

Let $X=0,1,2 $ with natural order topology and give $mathbbN$ its natural order topology. Consider $X times mathbbN$ with the dictionary order topology.

What are the limit points of $X times mathbbN$ ?

Answer:

$X times mathbbN=(0,1), (0,2), cdots, (0, infty), (1,1),(1,2), cdots, (1, infty), (2,1), (2,2), cdots, (2, infty) , cdots $

But how to find the limit point?

help me

limits order-topology

share|cite|improve this question

asked Mar 21 at 7:01

M. A. SARKARM. A. SARKAR

2,4621820

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Have you tried working with ordinals? Because that would immediately answer this question.
  $endgroup$
  – Don Thousand
  Mar 21 at 7:13
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @DonThousand, Any finite set has all its limits points. But this est $X times mathbbN$ is discrete set, so it is closed and hence all limits points. Is it?
  $endgroup$
  – M. A. SARKAR
  Mar 21 at 7:42
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  I wouldn't be so sure about that. Consider $(3,1)$. Any neighborhood around $(3,1)$ must contain an element of the form $(2,n)$ for some $ninmathbb N$. So, $(3,1)$ is a limit point of $Xtimesmathbb N$. Again, thinking about this in terms of ordinals makes this problem a lot easier IMO.
  $endgroup$
  – Don Thousand
  Mar 21 at 7:55
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @DonThousand, what are the open sets look like in $ X times mathbbN$?
  $endgroup$
  – M. A. SARKAR
  Mar 21 at 7:59
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Order topology...google is your best friend.
  $endgroup$
  – Don Thousand
  Mar 21 at 7:59
  
  

  
  
  
  

 | 
show 2 more comments

0

$begingroup$

Let $X=0,1,2 $ with natural order topology and give $mathbbN$ its natural order topology. Consider $X times mathbbN$ with the dictionary order topology.

What are the limit points of $X times mathbbN$ ?

Answer:

$X times mathbbN=(0,1), (0,2), cdots, (0, infty), (1,1),(1,2), cdots, (1, infty), (2,1), (2,2), cdots, (2, infty) , cdots $

But how to find the limit point?

help me

limits order-topology

share|cite|improve this question

asked Mar 21 at 7:01

M. A. SARKARM. A. SARKAR

2,4621820

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Have you tried working with ordinals? Because that would immediately answer this question.
  $endgroup$
  – Don Thousand
  Mar 21 at 7:13
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @DonThousand, Any finite set has all its limits points. But this est $X times mathbbN$ is discrete set, so it is closed and hence all limits points. Is it?
  $endgroup$
  – M. A. SARKAR
  Mar 21 at 7:42
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  I wouldn't be so sure about that. Consider $(3,1)$. Any neighborhood around $(3,1)$ must contain an element of the form $(2,n)$ for some $ninmathbb N$. So, $(3,1)$ is a limit point of $Xtimesmathbb N$. Again, thinking about this in terms of ordinals makes this problem a lot easier IMO.
  $endgroup$
  – Don Thousand
  Mar 21 at 7:55
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @DonThousand, what are the open sets look like in $ X times mathbbN$?
  $endgroup$
  – M. A. SARKAR
  Mar 21 at 7:59
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Order topology...google is your best friend.
  $endgroup$
  – Don Thousand
  Mar 21 at 7:59
  
  

  
  
  
  

 | 
show 2 more comments

$begingroup$

Let $X=0,1,2 $ with natural order topology and give $mathbbN$ its natural order topology. Consider $X times mathbbN$ with the dictionary order topology.

What are the limit points of $X times mathbbN$ ?

Answer:

$X times mathbbN=(0,1), (0,2), cdots, (0, infty), (1,1),(1,2), cdots, (1, infty), (2,1), (2,2), cdots, (2, infty) , cdots $

But how to find the limit point?

help me

limits order-topology

share|cite|improve this question

asked Mar 21 at 7:01

M. A. SARKARM. A. SARKAR

2,4621820

$endgroup$

share|cite|improve this question

asked Mar 21 at 7:01

M. A. SARKARM. A. SARKAR

2,4621820

share|cite|improve this question

asked Mar 21 at 7:01

M. A. SARKARM. A. SARKAR

2,4621820

asked Mar 21 at 7:01

M. A. SARKARM. A. SARKAR

2,4621820

asked Mar 21 at 7:01

M. A. SARKARM. A. SARKAR

2,4621820