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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)

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    The product of finitely many compact sets is compact, without any appeal to choice.2012-10-24
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    @Asaf It's off the topic, but can Compactness be defined without mentioning metric? I mean, say, $A$ is a compact set in a metric space $(X,d)$. Then it might be not compact in $(X,d')$?2012-10-24
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    Compactness is a *topological* construct. Namely, every open cover has a finite subcover. There are, of course, several ways to define compactness and they are equivalent under AC. In fact, the assertion that some of these forms *are equivalent* already imply AC. However compactness in its plain form is about open covers, and no compactness is related to metric at all. So if the two metrics are *equivalent topologically* the compactness is always preserved.2012-10-24
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    @Katlus: $[0,1]$ is compact with standard mretic, but not with discrete metric, if that's what you are after.2012-10-24
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    [compact Hausdorff space and continuity](http://math.stackexchange.com/q/134024)2012-10-24
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    Have you seen my answer below?2012-10-28
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    @Camilo I just saw your answer. Thanks for the posting:)2012-10-28

2 Answers 2

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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 $A\times B$.

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I think Rudin's exercise asks about a real function of a real variable, $f:E\to\mathbb R$, where $E\subseteq\mathbb R$. Graph of such a function is a subset of the plane $\mathbb R^2$, on which a metric is defined by $$d((x_1,y_1),(x_2,y_2))=\sqrt{(x_1-x_2)^2+(y_1-y_2)^2}$$ (see Example 2.16 in Rudin's book.)

For general metric spaces you can define many different product metrics, generalizing the result.