1st part of Exercise #27 is:(adsbygoogle = window.adsbygoogle || []).push({});

Define a point p in a metric space X to be a condensation point of a set E in X if every neighborhood of p contains uncountably many points of E. Suppose E is in R^k, E is uncountable and let P be the set of all condensation points of E. Prove P is perfect.

Obviously, P is closed. But I cannot figure out why every point of P is a limit point of P. There are several supposed solutions to this in the internet, but each of them only shows that if x is in P, x is a limit point of E(which is obvious since x is a condensation point of E). But the exercise asks us to prove that P is perfect, so if x is in P, x must be a limit point of P right? Could anyone offer any guidance to what such a proof would look like?

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# Baby rudin condensation points

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