# Show a set is compact

• Jan 22nd 2011, 08:46 AM
billa
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.
• Jan 22nd 2011, 08:51 AM
roninpro
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 $\displaystyle \pi_2(x,y)=y$.
• Jan 22nd 2011, 09:02 AM
DrSteve
By the way, I think you mean "cartesian product" and not "cross product"

I think that both methods suggested by roninpro should work.
• Jan 22nd 2011, 10:32 AM
Tinyboss
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.