Question

every collection of disjoint open intervals is countable

Answer 1

we know that rations are dense in the reals

Answer 2

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

Answer 3

disjoint open intervals*

Answer 4

i need to prove this

Answer 5

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

Answer 6

http://math.stackexchange.com/questions/318299/any-open-subset-of-bbb-r-is-a-at-most-countable-union-of-disjoint-open-interv

Answer 7

I can help if you have questions on any part.

Answer 8

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.

Answer 9

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

Answer 10

@zzr0ck3r is that interval \( I\) open

Answer 11

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)

Answer 12

for any x

Answer 13

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

Answer 14

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

Answer 15

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.

Answer 16

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) ?

Answer 17

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

Answer 18

aight ill check back later. movie time with the wifey

Answer 19

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.

Answer 20

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

Answer 21

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