Open Sets of R^n

  • Thread starter qspeechc
  • Start date
  • #1
836
13

Main Question or Discussion Point

Hi

Ok, so I know open sets of the real line are the countable union of disjoint open intervals (or open balls). Does this in any way extent to R^n? Say, any open set in R^n is the countable union of open rectangles or balls? I ask because I was reading some proof, and at a crucial step they use the fact that open sets in R^n can be expressed as the countable union of open rectangles, and I have no idea where this comes from! It doesn't even seem plausible to me, if one considers the open ball in the plane-- how would you describe that as the union of countably many open rectangles?
I know every open set is the union of open balls, and maybe the Heine-Borel theorem comes in somewhere...but I'm just lost.
Any help?
Thanks.
 

Answers and Replies

  • #2
tiny-tim
Science Advisor
Homework Helper
25,832
249
I ask because I was reading some proof, and at a crucial step they use the fact that open sets in R^n can be expressed as the countable union of open rectangles, and I have no idea where this comes from! It doesn't even seem plausible to me, if one considers the open ball in the plane-- how would you describe that as the union of countably many open rectangles?
Hi qspeechc! :smile:

What's uncountable about it?

n times countable is still countable. :wink:

For your open ball, just keep halving the size of the rectangles … that'll do the job, won't it? :smile:
 
  • #3
836
13
Errr...I'm not sure I follow you...? Are you saying I can go from the case of the real line to R^n with no difficulty?
 
  • #4
tiny-tim
Science Advisor
Homework Helper
25,832
249
Errr...I'm not sure I follow you...? Are you saying I can go from the case of the real line to R^n with no difficulty?
Yes. :smile:
 
  • #5
836
13
Hhmm, actually I found a proof, and it has nothing to do with the case of the real line, lol. It has to do with the fact that the rationals are dense in the reals.
 

Related Threads for: Open Sets of R^n

Replies
1
Views
4K
  • Last Post
Replies
18
Views
4K
Replies
18
Views
3K
  • Last Post
Replies
5
Views
1K
Replies
8
Views
2K
  • Last Post
Replies
3
Views
3K
  • Last Post
Replies
11
Views
2K
Replies
4
Views
2K
Top