June 3rd, 2018, 10:53 PM  #1 
Member Joined: Apr 2017 From: India Posts: 34 Thanks: 0  Real Analysis
Do the closed set consists of points other than limit points? If yes, then what are the examples? Last edited by skipjack; June 5th, 2018 at 07:30 AM. 
June 4th, 2018, 01:33 PM  #2 
Global Moderator Joined: May 2007 Posts: 6,613 Thanks: 617 
Isolated points?

June 5th, 2018, 06:58 AM  #3 
Member Joined: Apr 2017 From: India Posts: 34 Thanks: 0 
Yes points that are apart from limit points in the set. I didn't get what you meant by Isolated points.

June 5th, 2018, 07:23 AM  #4 
Senior Member Joined: Aug 2017 From: United Kingdom Posts: 270 Thanks: 81 Math Focus: Algebraic Number Theory, Arithmetic Geometry  Every point $x$ of a set is a limit point of that set (even if it's an "isolated point"). Indeed, consider the constant sequence $x, x, \dots$.

June 5th, 2018, 07:51 AM  #5 
Senior Member Joined: Aug 2012 Posts: 2,045 Thanks: 584  The set $[0,1] \cup \{2\}$ is closed but $2$ is not a limit point. It's an isolated point. It has a neighborhood that contains no other point of the set besides itself.

June 5th, 2018, 11:14 AM  #6 
Senior Member Joined: Aug 2017 From: United Kingdom Posts: 270 Thanks: 81 Math Focus: Algebraic Number Theory, Arithmetic Geometry 
Ah yes, ignore my previous post. I mistakenly took $x$ being a limit point to mean "$x$ is the limit of a sequence of points in the set" (rather than "$x$ is the limit of an eventually nonconstant sequence of points in the set"/"every neighbourhood of $x$ contains a point of the set except $x$).

June 5th, 2018, 11:54 AM  #7  
Senior Member Joined: Aug 2012 Posts: 2,045 Thanks: 584  Quote:
An adherent point, also known as a point of closure, is a point whose every neighborhood contains some point of the set. So $2$ is a point of closure of $[0,1] \cup \{2\}$. But it's not a limit point, which requires that every neighborhood of the point contains some point of the set other than the point in question.  
June 5th, 2018, 02:23 PM  #8  
Senior Member Joined: Aug 2017 From: United Kingdom Posts: 270 Thanks: 81 Math Focus: Algebraic Number Theory, Arithmetic Geometry  Quote:
 

Tags 
analysis, closed set, limit points, real 
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