Consider two metric spaces, X and Y with X being compact. Let f:X --> Y and define graph(f)=set of all points (x, f(x)) with x inside X. Then f is continuous iff it's graph is compact.(adsbygoogle = window.adsbygoogle || []).push({});

Well my main question is how do we make sense of open sets in the space in which the graph is contained? I'm aware that since because this is a finite product the product topology is equivalent to the inherited topology on the product (that is defining an open set to be the product of open sets in each X and Y) - that being said my book (Rudin) doesn't go that far in fact he doesn't MENTION metric spaces that are products of other metric spaces (except for n-space of course) Can I just assume a finite Cartesian product of compact sets is compact? Or does Rudin want me to solve it for euclidean spaces only?!

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# Graph of a Continous function

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