Prove that any family of pairwise disjoint open intervals is countable.

Question

Having trouble solving the following problem:

Any help would be great!

http://math.stackexchange.com/questions/5049/countability-of-disjoint-intervals — njguliyev, Oct 14 '13 at 14:55

DonAntonio · Answer 1 · 2013-10-14T15:03:00.147

2

Hint:

In every such open (non-trivial) interval there's at least one rational, and the union of all these rationals over that family is at most $\;\Bbb Q\;$ , and the cardinality of $\;\Bbb Q\; $ is....

Or another (kind of) approach: as before with the rationals: if the family were uncountable, then taking the union of it would yield uncountable rationals...

edited Oct 14 '13 at 15:03

answered Oct 14 '13 at 14:56

DonAntonio

214,715

1

Awesome! Thank you very much for your help. – Jill Oct 14 '13 at 15:07

Prove that any family of pairwise disjoint open intervals is countable.

1 Answers1

Linked

Related