Open Sets of R^n: Countable Union of Open Rectangles/Balls?

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In summary: So if you want to cover an interval with rectangles, you can do so without having to overlap any of the intervals. It's a bit of a stretch, but that's what they use to prove their theorem. In summary, this proof relies on the fact that open sets in the real line can be expressed as the countable union of open rectangles.
  • #1
qspeechc
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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.
 
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  • #2
qspeechc said:
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
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
qspeechc said:
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
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 to Open Sets of R^n: Countable Union of Open Rectangles/Balls?

What is an open set in R^n?

An open set in R^n is a set of points where every point has a neighborhood contained entirely within the set. This means that there is always a small enough radius, or distance, around each point where all the points within that radius are also included in the set.

What is a countable union of open rectangles/balls?

A countable union of open rectangles/balls is a collection of open rectangles or balls that can be listed or counted. This means that there is a finite or infinite number of open rectangles or balls that can be combined to form the set.

How do open sets and countable unions relate to each other?

An open set can be represented as a countable union of open rectangles or balls. This means that the open set can be broken down into smaller, simpler components that are combined together to form the set.

Why is it useful to represent open sets as countable unions?

Representing open sets as countable unions allows for easier understanding and manipulation of these sets. It also allows for the use of certain mathematical theorems and tools that can be applied to countable unions, making it a useful representation for analysis and proofs.

What are some real-world applications of open sets and countable unions?

Open sets and countable unions are used in many fields of science and engineering, including physics, computer science, and economics. For example, in physics, open sets are used to represent regions in space where particles can move freely, and countable unions are used to analyze and model complex systems. In computer science, open sets are used in data structures and algorithms to efficiently store and search for information. In economics, open sets and countable unions are used to study market trends and make predictions about future behavior.

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