# Intersection of compact sets

• Mar 10th 2010, 05:23 PM
harish21
Intersection of compact sets
Given, two compact sets A and B, how do we show that their intersection A$\displaystyle \bigcap$B is also compact?
• Mar 10th 2010, 05:52 PM
Drexel28
Quote:

Originally Posted by harish21
Given, two compact sets A and B, how do we show that their intersection A$\displaystyle \bigcap$B is also compact?

In what kind of space?
• Mar 10th 2010, 05:59 PM
harish21
Quote:

Originally Posted by Drexel28
In what kind of space?

A and B are compact subsets of R
• Mar 10th 2010, 06:01 PM
Drexel28
Quote:

Originally Posted by harish21
A and B are compact subsets of R

Think closed subspace
• Mar 10th 2010, 06:07 PM
harish21
Quote:

Originally Posted by Drexel28
Think closed subspace

A and B are closed and bounded because they are compact.
So A $\displaystyle \bigcap$ B is also closed.
How do we show boundedness?
• Mar 10th 2010, 06:24 PM
Drexel28
Quote:

Originally Posted by harish21
A and B are closed and bounded because they are compact.
So A $\displaystyle \bigcap$ B is also closed.
How do we show boundedness?

The Heine-Borel theorem is kind of overkill here...but.

Clearly if $\displaystyle C\subseteq D$ then $\displaystyle \text{diam }C\leqslant \text{diam }D$, right? $\displaystyle A\cap B\subseteq A$