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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 $G=\{(x,f(x))\in X\times Y:x\in E\}$.

- Proving absolute value inequalities
- Properties of function $f(x) = (1 + x^2)^{-\alpha/2}(\log(2+x^2))^{-1},\text{ }x \in \mathbb{R}$ with $0 < \alpha < 1$.
- An alternating series …
- A continuous map that fixes the boundary of a domain pointwise is surjective
- The extension of smooth function
- Limit of sequence of continued fractions

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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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 $A\times B$.

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.

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