My Math Forum Is the union of infinite disjoint closed sets closed?

 Real Analysis Real Analysis Math Forum

 November 12th, 2012, 08:07 AM #1 Newbie   Joined: Nov 2012 Posts: 10 Thanks: 0 Is the union of infinite disjoint closed sets closed? Is the union of infinite closed sets (disconnected each other in given topology) closed? (remark: disconnected each other in given topology, that is, for any two such closed sets A and B there exists two open sets U and V such that U includes A and V includes B, and the intersection of U and V is an emptyset)
 November 12th, 2012, 09:44 AM #2 Senior Member   Joined: May 2008 From: York, UK Posts: 1,300 Thanks: 0 Re: Is the union of infinite disjoint closed sets closed? Do you mean countably infinite or uncountably infinite? (As it happens the answer is the same in both cases but it's considerably easier to see why in the uncountable case).
November 12th, 2012, 03:29 PM   #3
Global Moderator

Joined: May 2007

Posts: 6,660
Thanks: 648

Re: Is the union of infinite disjoint closed sets closed?

Quote:
 Originally Posted by 03sqq Is the union of infinite closed sets (disconnected each other in given topology) closed? (remark: disconnected each other in given topology, that is, for any two such closed sets A and B there exists two open sets U and V such that U includes A and V includes B, and the intersection of U and V is an emptyset)
Not necessarily. In the usual topology of reals, single points are closed sets. However, the set of all rationals is not a closed set.

 November 12th, 2012, 03:33 PM #4 Global Moderator   Joined: May 2007 Posts: 6,660 Thanks: 648 Re: Is the union of infinite disjoint closed sets closed? If you want a better set meeting your condition, use the set of points {1/n}. These can be separated the way you described, but the union is not a closed set.
 November 13th, 2012, 04:40 AM #5 Newbie   Joined: Nov 2012 Posts: 10 Thanks: 0 Re: Is the union of infinite disjoint closed sets closed? Thanks a lot for all replies, especially thanks mathman for the simple and sagacious answer.

 Tags closed, disjoint, infinite, sets, union

,

,

,

,

,

,

,

,

,

,

,

,

# disjoint union closed sets

Click on a term to search for related topics.
 Thread Tools Display Modes Linear Mode

 Similar Threads Thread Thread Starter Forum Replies Last Post veronicak5678 Real Analysis 4 April 19th, 2012 05:37 PM mizunoami Real Analysis 3 November 30th, 2011 07:06 PM brangelito Number Theory 8 September 28th, 2011 08:11 AM farzyness Real Analysis 1 February 18th, 2009 04:58 PM tmac3813 Real Analysis 2 October 9th, 2008 07:20 PM

 Contact - Home - Forums - Cryptocurrency Forum - Top