Is an Open Set in a Metric Space a Countable Union of Disjoint Open Intervals?

[ehrenfest]

Homework Statement

Let R be the metric space of all real numbers. Prove that any bounded open set in R is a countable union of disjoint open intervals.

Homework Equations

The Attempt at a Solution

If the bounded open set is continuous (is continuity defined for set?), then it is itself an open interval. So we let it be multiple continuous, intervals. Then each of the of those intervals is open, I think?

 

[morphism]

When you say "continuous", do you mean "contains no gaps"? If so, then you have the right idea. All you have to do is formalize your argument.

Also, don't forget that you want to prove that the union is countable.

 

[ehrenfest]

When you say "continuous", do you mean "contains no gaps"? If so, then you have the right idea. All you have to do is formalize your argument. Also, don't forget that you want to prove that the union is countable.

Yes, let's say the set X is "gapless" if for every two points in X we can find an open ball that contains those two points and all other points it contains are also in X.

So if the open set Y is gapless, we can express it as an open interval. If not I will need to show that it can be divided into a countable number of gapless sets.

So, the way to divide an open set in R into a disjoint union of gapless sets would be to find every two points between which a gap exists and separate them into different sets. I think there cannot be an infinite number of finite gaps or else they would take up all of R, but they may be wrong?