Open subsets are a union of disjoint open intervals

In summary, it is possible to write any open subset of \Real as an at most countable union of disjoint open intervals. This can be shown by defining an equivalence relation and using the archimedean property to find a countable number of intervals that cover the set.
  • #1
redone632
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Homework Statement


Prove that any open subset of [itex]\Real[/itex] can be written as an at most countable union of disjoint open intervals.


Homework Equations


An at most countable set is either finite or infinitely countable.


The Attempt at a Solution


It seems very intuitive but I am at lost where to even start. We're doing compactness in metric spaces so I would assume it must apply. But I thought a set has to be closed in order to be compact and this deals with an open subset so it can't possibly be compact. Any help would be much appreciated!
 
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  • #2
This has pretty much nothing to do with compactness. Try this:
1) Show you can write it as a union of disjoint intervals.
2) Think about rational numbers
 
  • #3
Hmmm, I think I've made some development but I'm still unsure how solid it is.

Let [itex]E[/itex] be an open subset of [itex]\Real[/itex]. Define the equivalence relation [itex]x \sim y \Longleftrightarrow \exists (a,b)[/itex] such that [itex]{x,y} \in (a,b) \subseteq E[/itex]. The equivalence classes will be distinct. By the archimedean property, there exists a distinct rational number in each interval. Since the rationals are countable, we have a countable number of intervals. The countable union of these intervals will equal the set E. Therefore, every subset of R can be represented by a countable union of disjoint open intervals.
 

What does it mean for a subset to be "open"?

In mathematics, an open subset is a subset of a topological space that does not contain its boundary points. In other words, every point in an open subset has a neighborhood that is entirely contained within the subset.

What are "disjoint open intervals"?

Disjoint open intervals are intervals on the real number line that do not share any common points. In other words, there is no overlap between the intervals.

How do you know an open subset can be written as a union of disjoint open intervals?

This is a fundamental property of topological spaces. In particular, it is a consequence of the definition of open sets and the topology of the space. Essentially, every open subset can be broken down into smaller, non-overlapping parts.

Why is it important for open subsets to be a union of disjoint open intervals?

This property allows for a more precise and efficient way to describe and analyze open subsets. By breaking them down into smaller intervals, we can better understand their properties and relationships with other subsets and the overall space.

Can an open subset be expressed as a union of non-disjoint intervals?

No, an open subset cannot be written as a union of non-disjoint intervals. This is because the definition of an open subset specifically excludes boundary points, so there can be no overlap between the intervals. If there is overlap, then the subset would not be considered open.

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