Proving Disjointness of Open Intervals in E Subset of R

[mynameisfunk]

Hey guys, doing another rudin-related question. Here Goes:

Show that if E \subseteq \Re is open, then E can be written as an at most countable union of disjoint open intervals, i.e., E=\bigcup_n(a_n,b_n). (It's possible that a_n=-\infty b_n=+\infty for some n.)

My attempt:
Take the set of all Neighborhoods of all of the rationals of a rational radius in R to be A. Now all members of E intersect A make up E. Take the union of all of the neighborhoods in this set E intersect A and this is a countable union of disjoint sets.

Is there a problem with this?

 

[mynameisfunk]

I now realize this doesn't create a disjoint union. I had the idea of taking an arbitrarily large neighborhood of a point in E that is still a proper subset of E and then filling up the gaps on either side with more neighborhoods, always still contained in E, until I have a countably dense set of neighborhoods contained in E...

Would this do?

 

[kntsy]

Hey guys, doing another rudin-related question. Here Goes:

Show that if E \subseteq \Re is open, then E can be written as an at most countable union of disjoint open intervals, i.e., E=\bigcup_n(a_n,b_n). (It's possible that a_n=-\infty b_n=+\infty for some n.)

My attempt:
Take the set of all Neighborhoods of all of the rationals of a rational radius in R to be A. Now all members of E intersect A make up E. Take the union of all of the neighborhoods in this set E intersect A and this is a countable union of disjoint sets.

Is there a problem with this?

Here is my suggestion:
First let E be a union of disjoint open sets F. Then E is an open set. We know that by Lindelof there is a countable subset F' of F. However, since F is disjoint, F=F'. So F is countable.

Accurately, F is a collection of sets and E is the union of sets taken from F.

I guess that your first suggestion proves the lindelof that countable subets exist. Although they are "not disjoint", you have proven a lindelof and you can apply this property to the open set which is a union "disjoint" open sets.

 

[Landau]

This is a familiar exercise/result. See e.g. here, here, and your own (!) thread here.