Q:

Continuity, Compactness and Graph

PMA, Rudin p.99 Exercise 6
Let $X,Y$ be metric spaces and $E$ be a compact subset of $X$.
Define $f:E\rightarrow Y$ and $G=\{(x,f(x))\in X\times Y:x\in E\}$.
Then prove that $f$ is continuous on $E$ iff $G$ is compact.
I'm not sure hypotheses Rudin made are sufficient to prove this. How do I know what kind of metric is in $X\times Y$? Is there a generally used metric of Cartesian product of two metric spaces, when metric of the product is not mentioned?
Next, say metric in $X\times Y$ is defined. Let $A,B$ be compact sets in $X,Y$ respectively. How do I prove that $A\times B$ is compact? I think this is inevitable in the proof for above theorem, but there was nothing about this in this book.. (I know generalization of this is Tychonoff's Theorem which needs choice)

A:

If $(X,d_1)$ and $(Y-d_2)$ are metric spaces, then $d((x,y),(x'y'))=d_1(x,y')+d_2(y,y')$ is a metric on $X\times Y$.
Note that this does not give the standard metric on e.g. $\mathbb R^2=\mathbb R \times\mathbb R$. But the topologies defined by both metrics are the same.
If $A,B$ are compact and we are given an open cover of $A\times B$, then for each $a\in A$ we find a finite subcover of $\{a\}\times B$. By using compactness of $B$, show that there is some $r>0$ such that for each $b\in B$, the ball $B_r(a,b)$ around $(a,b)$ with respect to metric $d$ on $X\times Y$ is in one of these finitely many covering sets. Thus thes finitly many open sets cover not just $\{a\}\times B$ but in fact $B_r(a)\times B$ (here the ball is with respect to metric $d_1$ on $X$). With varying $a$, the $B_r(a)$ cver $A$, hence there is a finite subcover, corresponsing to a finite subcover of $A\times B$.

