# Compact in R

• Dec 13th 2012, 10:31 AM
DrNerj
Compact in R
Hello!

I was wondering if someone could clear something up for me..

Is every non-empty, compact subset of R a closed interval in R?
It seems to me that this is true. I know that such sets are closed an bounded, but I don't know how to show that they're connected.

Any help is much appreciated!
• Dec 13th 2012, 11:03 AM
emakarov
Re: Compact in R
According to the Bolzano–Weierstrass theorem, a subset of $\mathbb{R}^n$ is compact if and only if it is closed and bounded. So, the union of two disjoint finite closed intervals in $\mathbb{R}$ is compact.
• Dec 13th 2012, 11:08 AM
Plato
Re: Compact in R
Quote:

Originally Posted by DrNerj
Is every non-empty, compact subset of R a closed interval in R?

$[0,1]\cup [2,3]$ is a non-empty, compact subset of $\mathbb{R}~.$

Any non-empty closed and bound subset of $\mathbb{R}$ is compact.
• Dec 13th 2012, 11:47 AM
DrNerj
Re: Compact in R
Thanks a lot!
That really helped clear things up.