# Thread: Intersection of compact sets

1. ## 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?

2. 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?

3. Originally Posted by Drexel28
In what kind of space?

A and B are compact subsets of R

4. Originally Posted by harish21
A and B are compact subsets of R
Think closed subspace

5. 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?

6. 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? $\displaystyle A\cap B\subseteq A$