Closed Intervals of R: Uncountable Collection Example

[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 doesn't 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.

 

[StatusX]

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

 

[barksdalemc]

StatusX,

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

 

[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.