Open set is countable union of disjoint open balls

In summary, in R, every nonempty open set is the disjoint union of a countable collection of open intervals, which can be proven by considering the largest open interval contained in the open set for each point in the set. However, for this proof to hold in general topological spaces, we also need the conditions of separability and locally path connectedness.
  • #1
dreamtheater
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In R, every nonempty open set is the disjoint union of a countable collection of open intervals. (Royden/Fitzpatrick, 4th edition)

What is the most general setting in which every open set is a disjoint union of countable collection of open balls (or bases)? In R^n? In metric spaces? In second countable topological spaces?

The above theorem in R took me by surprise when I saw it for the first time.
 
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  • #2
What is the proof of this fact in R? Let U be some open set in R. For every x in U, let I_x be the largest open interval containing x and contained entirely in U. Then for x and y in U, either I_x and I_y are equal or disjoint. Moreover, if I_x and I_y are disjoints, then by density of Q in R, they both contain a rational number that the other does not contain. So the set [itex]\{I_x:x\in U\}[/itex] is countable and its elements are disjoint open intervals whose union is U. q.e.d.

(By the way, notice that I_x is just the connected component of U containing x.)

In light of this, it seems to me that the only ingredient necessarily is separability:

Let X be a separable topological space. This means X admits a dense countable subset D. Let U be open in X. For each x in X, let C_x be the connected component of U containing x. Then for x and y in U, either C_x and C_y are equal or disjoint. Moreover, if C_x and C_y are disjoints, then by density of D] in X, they both contain an element of D that the other does not contain. So the set [itex]\{C_x:x\in U\}[/itex] is countable and its elements are disjoint open subsets of X whose union is U.
 
  • #3
Ah, no. For this proof, we also need to add on X the hypothesis that the connected components of U will be open. This will be the case for instance if X is locally path connected.

Indeed, consider the following counter-example. Let Q have the subspace topology inherited by R. Let r<s be any two irrational numbers. Then [itex]U:=]r,s[\cap \mathbb{Q}[/itex] is open in Q but its connected components are all the singletons {q} with q rational and contained in ]r,s[. By by density of the irrationals in R, none of the {q} are open in Q.
 

Related to Open set is countable union of disjoint open balls

What does it mean for an open set to be a countable union of disjoint open balls?

This means that the open set can be expressed as a collection of open balls that do not intersect with each other, and the number of open balls in this collection is countable (finite or infinite).

Why is it important to understand this concept?

Understanding this concept is important because it helps in the study of topology and analysis, where open sets play a crucial role. It also allows us to express complicated open sets in a simpler form, making it easier to work with them.

How do we prove that an open set is a countable union of disjoint open balls?

This can be proven using the concept of open covers. We start by covering the open set with a collection of open balls. Then, we show that this covering can be reduced to a countable collection of disjoint open balls that still cover the original open set.

Can an open set be expressed as an uncountable union of disjoint open balls?

No, an open set can only be expressed as a countable union of disjoint open balls. This is because an open set can only be covered by a countable number of open balls, and if we allow an uncountable union, there will always be points in the open set that are not covered by any of the open balls.

How does this concept relate to the concept of open coverings?

The concept of open set being a countable union of disjoint open balls is closely related to the concept of open coverings. An open set being a countable union of disjoint open balls is just one way of expressing an open set as an open covering. It is a more specific and simpler form of open coverings.

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