anonymous one year ago every collection of disjoint open intervals is countable

1. anonymous

we know that rations are dense in the reals

2. zzr0ck3r

This is true for any open set in R. It as most a countable union of open intervals

3. zzr0ck3r

disjoint open intervals*

4. anonymous

i need to prove this

5. zzr0ck3r

I am sure if you google it you will find many proofs. Anything I do will be a regurgitation.

6. zzr0ck3r
7. zzr0ck3r

I can help if you have questions on any part.

8. zzr0ck3r

if $$O$$ is an open set and $$x∈O$$ then there exists an interval I such that $$x∈I⊂O$$. If there exists one such interval, then there exists one 'largest' interval which contains $$x$$ (the union of all such intervals). Denote by $$\{I_α\}$$ the family of all such maximal intervals. First all intervals $$I_α$$ are pairwise disjoint (otherwise they wouldn't be maximal) and every interval contains a rational number, and therefore there can only be a countable number of intervals in the family.

9. zzr0ck3r

If you are more in to set theory then there is a pretty cool proof using that approach.

10. anonymous

@zzr0ck3r is that interval $$I$$ open

11. anonymous

for example, (0,1) is open because if x is inside (0,1) you can form a 'ball' around x that is contained in (0,1)

12. anonymous

for any x

13. zzr0ck3r

Yes, this is the definition of open set on order topology.

14. zzr0ck3r

technically (0,1) is a basis element for the topology, but you got the right idea.

15. zzr0ck3r

basically because any collection of intervals will contain a rational point, we can index the intervals with one such rational point. The rational numbers are countable, so the collection is a countable collection.

16. anonymous

so for example the point x= 0.25 in (0,1) the maximal open interval that is still contained in (0,1) would be (0, .5) ?

17. zzr0ck3r

I would say more like $0.25\in (0,1)\cup(34,199)$ then we would get $$(0,1)$$

18. zzr0ck3r

aight ill check back later. movie time with the wifey

19. anonymous

I may be missing a condition here, but I was assuming the maximal interval had to be symmetric about x. So for $$x = 0.25$$ and $$O =(0,1)\cup(34,199)$$, if you say that the interval $$I$$ does not have to be symmetric about $$x$$, then wouldn't the whole interval $$(0,1)\cup(34,199)$$ would be the maximal set that contains $$x$$ ? Thanks for discussing this, i am trying to learn this.

20. zzr0ck3r

no there is no need for it to be symmetric about x

21. zzr0ck3r

we want a maxl interval that contains x, not a maxl open set that contains x