
Real Analysis Real Analysis Math Forum 
 LinkBack  Thread Tools  Display Modes 
February 3rd, 2013, 10:58 AM  #1 
Senior Member Joined: Apr 2012 Posts: 106 Thanks: 0  Closed set and closed ball
Here is the claim: If set A is closed, then there exists a family of sets D, whose elements are closed balls, so that Here is another claim: If set A is open, then there exist family D, whose elements are open balls, to that Here is a try: i take X that is in . X is also in open D_n so it's neighbourghood is also in D_n. D_n is a part of , therefore neighbourghood of X is also in . Which means that open balls form an open set. I am not sure if this is a right way of thinking. What if it's reverse. So what happens if I take that x is in A? 
February 3rd, 2013, 01:24 PM  #2 
Global Moderator Joined: May 2007 Posts: 6,660 Thanks: 648  Re: Closed set and closed ball
The statement about open sets is correct and your proof is valid. The closed set statement is obviously false. Consider a closed set consisting of just two points. An intersection of closed balls will be connected or empty. 
February 3rd, 2013, 01:33 PM  #3  
Senior Member Joined: Apr 2012 Posts: 106 Thanks: 0  Re: Closed set and closed ball Quote:
 
February 4th, 2013, 01:49 PM  #4  
Global Moderator Joined: May 2007 Posts: 6,660 Thanks: 648  Re: Closed set and closed ball Quote:
Consider 1d case. Closed balls are closed intervals. The intersection of two intervals is either empty or an interval. You can never get two separate pieces.  

Tags 
ball, closed, set 
Thread Tools  
Display Modes  

Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
Why is this set not closed?  E7.5  Calculus  2  February 25th, 2014 05:56 AM 
Closed Set  alejandrofigueroa  Real Analysis  3  September 17th, 2013 03:53 PM 
Closed Set  sebaflores  Real Analysis  2  September 17th, 2013 06:30 AM 
Is the union of infinite disjoint closed sets closed?  03sqq  Real Analysis  4  November 13th, 2012 04:40 AM 
Boundary of closed unit ball?  martexel  Real Analysis  1  December 6th, 2009 05:50 PM 