Homework Help: Analysis Question

1. Sep 23, 2006

barksdalemc

Guys I would appreciate any help on this. I've been trying to find an example of a collection of closed intervals of R that is uncountable. I proved that if I take a collection of open intervals of R and bijectively map them to Z, then the collection is countable, and I would assume the same with a collection of closed intervals, but clearly there must be an example where that doesnt happen and I don't understand why my logic on the collection of open sets cannot be extended to the collection of closed sets. Thanks for any help.

2. Sep 23, 2006

StatusX

The collection { [0,a] | a$\in$A}, where A is a subset of (0,$\infty$), can be put in a bijection with A.

3. Sep 23, 2006

barksdalemc

StatusX,

I forgot to mention the closed sets have to be disjoint.

4. Sep 23, 2006

StatusX

Well then you could always take points as your closed intervals. It is not possible to form an uncountable set of disjoint closed intervals, each of finite length.