Infinite Collection, compact sets

• Dec 17th 2009, 05:07 PM
p00ndawg
Infinite Collection, compact sets
Find an infinite collection {Sn : n in N} of compact sets in R such that infinite union of Sn is not compact.

Is it correct to say that the union of infinite sets is a countable infinity?
So that if the union of infinite sets is going to be bounded by N, but you could find a set N + 1 that is not in the set?
• Dec 17th 2009, 05:18 PM
platinumpimp68plus1
Well, if you set Sn=[-n,n], then take the union of each Sn for n=1,2,3,4...

Each [-n,n] is clearly compact. But the union (which is R itself), is not.
• Dec 17th 2009, 09:49 PM
Drexel28
Quote:

Originally Posted by p00ndawg

Is it correct to say that the union of infinite sets is a countable infinity?

Be careful here. If you have a set $\mathcal{N}=\left\{S_n:n\in\mathbb{N}\right\}$ then clearly $\mathcal{N}$ is countable for the "worst case scenario" is that all the sets are distinct in which case the mappking $f:\mathbb{N}\mapsto\mathcal{N}$ given by $f(n)=S_n$ is clearly a bijection. Saying that given a countable class of sets the union is automatically countable is just plain wrong. What about $\mathcal{N}=\left\{\mathbb{R}^n:n\in\mathbb{N}\rig ht\}$?