Open sets, countable unions of open rectangles

  • Thread starter infk
  • Start date
  • Tags
    Sets
In summary, the conversation discusses a proof about connected open sets and their relation to open, disjoint rectangles. The first part of the proof shows that if a set is a connected open set, it cannot be written as a countable union of open, disjoint rectangles. The second part of the proof shows that if a set can be written as a countable union of open, disjoint rectangles, it must be a rectangle. This is proven using contradiction and the lemma that a union of two or more disjoint open rectangles is not connected.
  • #1
infk
21
0

Homework Statement


So here is a "proof" from my measre theory class that I don't really understand. Be nice with me, this is the first time I am learning to "prove" things.
Show that a connected open set Ω ([itex]\mathbb{R}^d[/itex], I suppose) is a countable union of open, disjoint rectangles if and only if Ω is itself a rectangle.


Homework Equations


N/A

The Attempt at a Solution


Taught in class:
An open set Ω is connected if and only if it is impossble to write Ω = V [itex]\bigcup[/itex]U where U and V are open, non-empty and disjoint. Thus if we can write Ω = [itex]\bigcup^{\infty}_{k=1} R_k[/itex] where [itex]R_k[/itex] are open disjoint rectangles of which at least two are non-empty (lets say [itex]R_1[/itex] and [itex]R_2[/itex] ) we can then write Ω = [itex]R_1 \cup (\bigcup^{\infty}_{k=2} R_k) [/itex] and therefore Ω is not connected.
There is also another question; Show that an open disc in [itex]\mathbb{R}^2[/itex] is not a countable union of open disjoint rectangles. To show this, the professor said that the previous result apllies since a disc is connected.

I don't understand:
Why does this prove the proposition?, Is the assumption that we can write Ω = [itex]\bigcup^{\infty}_{k=1} R_k[/itex] where [itex]R_k[/itex] are open disjoint rectangles of which at least two are non-empty, equvalent with saying that Ω is a rectangle? If it is, have we not then assumed that Ω is a rectangle and then shown that a rectangle is not connected?
:confused:
 
Physics news on Phys.org
  • #2
First, if Ω is a rectangle, then we can write it as a "union of disjoint open rectangles" by taking the set of such rectangles to include only Ω itself.

To show the other way, that if Ω can be written as a "union of disjoint open rectangles then it is a rectangle", use proof by contradiction- if Ω is not a rectangle, then such a union cannot consist of a single rectangle. But the "lemma" you give shows that the union of two or more disjoint open rectangles is not connected, so cannot be a rectangle as all rectangles are connected. That gives a contradiction, proving the theorem.
 

FAQ: Open sets, countable unions of open rectangles

1. What are open sets?

Open sets are subsets of a metric space that contain all the points within the set's boundary. In other words, they do not include any of their boundary points.

2. What is a countable union of open rectangles?

A countable union of open rectangles is a collection of open rectangles that can be expressed as a union of a finite or infinite number of open rectangles. This means that the set can be written as a sum of open rectangles, each of which is an open set.

3. How are open sets and countable unions of open rectangles related?

Open sets and countable unions of open rectangles are related in that every open set can be expressed as a countable union of open rectangles. This is known as the Borel decomposition theorem.

4. What is the significance of open sets and countable unions of open rectangles in mathematics?

Open sets and countable unions of open rectangles are important concepts in mathematics, particularly in analysis and topology. They allow for the study of convergence and continuity of functions, as well as the development of important theorems such as the Hahn-Banach theorem and the Baire category theorem.

5. How are open sets and countable unions of open rectangles used in real-world applications?

Open sets and countable unions of open rectangles have various applications in fields such as physics, engineering, and computer science. They are used to model and analyze mathematical systems and can be applied to problems involving optimization, signal processing, and data analysis, among others.

Similar threads

Back
Top