My Math Forum  

Go Back   My Math Forum > College Math Forum > Real Analysis

Real Analysis Real Analysis Math Forum


Thanks Tree3Thanks
  • 1 Post By mathman
  • 1 Post By cjem
  • 1 Post By Maschke
Reply
 
LinkBack Thread Tools Display Modes
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.
shashank dwivedi is offline  
 
June 4th, 2018, 01:33 PM   #2
Global Moderator
 
Joined: May 2007

Posts: 6,558
Thanks: 602

Isolated points?
Thanks from romsek
mathman is offline  
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.
shashank dwivedi is offline  
June 5th, 2018, 07:23 AM   #4
Senior Member
 
Joined: Aug 2017
From: United Kingdom

Posts: 211
Thanks: 64

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 $x, x, \dots$.
cjem is offline  
June 5th, 2018, 07:51 AM   #5
Senior Member
 
Joined: Aug 2012

Posts: 1,971
Thanks: 550

Quote:
Originally Posted by shashank dwivedi View Post
Yes points that are apart from limit points in the set. I didn't get what you meant by Isolated points.
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.
Maschke is online now  
June 5th, 2018, 11:14 AM   #6
Senior Member
 
Joined: Aug 2017
From: United Kingdom

Posts: 211
Thanks: 64

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 non-constant sequence of points in the set"/"every neighbourhood of $x$ contains a point of the set except $x$).
Thanks from Joppy
cjem is offline  
June 5th, 2018, 11:54 AM   #7
Senior Member
 
Joined: Aug 2012

Posts: 1,971
Thanks: 550

Quote:
Originally Posted by cjem View Post
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 non-constant sequence of points in the set"/"every neighbourhood of $x$ contains a point of the set except $x$).
This is a tricky point (no pun) that confuses everyone.

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.
Thanks from Joppy
Maschke is online now  
June 5th, 2018, 02:23 PM   #8
Senior Member
 
Joined: Aug 2017
From: United Kingdom

Posts: 211
Thanks: 64

Math Focus: Algebraic Number Theory, Arithmetic Geometry
Quote:
Originally Posted by Maschke View Post
This is a tricky point (no pun) that confuses everyone.

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.
Yeah, I'm used to the terms "accumulation point" and "point of closure" for the two concepts. I knew limit point referred to one of the two, but settled on the wrong one without bothering to check!
cjem is offline  
Reply

  My Math Forum > College Math Forum > Real Analysis

Tags
analysis, closed set, limit points, real



Thread Tools
Display Modes


Similar Threads
Thread Thread Starter Forum Replies Last Post
help in real analysis saifns Real Analysis 2 November 18th, 2015 06:00 AM
Real Analysis. Luiz Real Analysis 1 May 9th, 2015 01:36 PM
Prove between Real Analysis and Complex Analysis uniquesailor Real Analysis 2 January 3rd, 2012 09:56 PM
real analysis eleni12344 Real Analysis 0 April 16th, 2009 12:14 PM
real analysis? clooneyisagenius Real Analysis 0 February 26th, 2008 03:35 PM





Copyright © 2018 My Math Forum. All rights reserved.