# Thread: Show a set is compact

1. ## Show a set is compact

If A and B are compact subsets of some metric space M with metric d, then I think that the cross product $\displaystyle A\times B$ is a compact subset of the metric space $\displaystyle M^2$ where the metric in this space is defined by

$\displaystyle p((x,y),(z,w))=\sqrt{(d(x,z)^2+d(y,w)^2}$

What is the best way to show that AxB is indeed compact? Is there a particularly good way to do this? I think I have a proof showing that every sequence in AxB has a convergent subsequence.

Thanks.

2. Showing that every sequence has a convergent subsequence is probably the easiest way to do it. You would need to use the fact that a sequence $\displaystyle \{(x_n,y_n)\}$ in $\displaystyle A\times B$ converges if and only if $\displaystyle \{x_n\}$ converges in $\displaystyle A$ and $\displaystyle \{y_n\}$ converges in $\displaystyle B$.

On the other hand, it might be possible to use the definition directly: take an open cover of $\displaystyle A\times B$ and try to show that it has a finite subcover. You can relate this to the compactness of $\displaystyle A$ and $\displaystyle B$ by using the projection maps: $\displaystyle \pi_1: A\times B\to A$ given by $\displaystyle \pi_1(x,y)=x$ and $\displaystyle \pi_2: A\times B\to B$ given by $\displaystyle \pi_2(x,y)=y$.

3. By the way, I think you mean "cartesian product" and not "cross product"

I think that both methods suggested by roninpro should work.

4. Show that the given metric induces the product topology, and invoke the theorem that finite (easy proof) or arbitrary (Tychonoff) products of compact spaces are compact.