Subsets of compact spaces

• Apr 17th 2013, 01:34 PM
I-Think
Subsets of compact spaces
Can anyone provide me with an example of a compact space that has subsets that aren't compact?
• Apr 17th 2013, 02:00 PM
Plato
Re: Subsets of compact spaces
Quote:

Originally Posted by I-Think
Can anyone provide me with an example of a compact space that has subsets that aren't compact?

Any compact subset of a Hausdorff space with the relative topology is a compact space.
Now any open set is that compact subspace is not compact. WHY?

Example: $[-2,2]\subset \mathbb{R}^1$ with the relative topology is a compact space.

But $(0,1)$ is not compact.