# How to prove that the set of condensation points of an uncountable subset of the real euclidean k-space is perfect?

I'm referring to Problem 27 in the exercises of Chapter 2 in the textbook, Principles of Mathematical Analysis, 3rd edition, by Walter Rudin.

I've managed to prove that the set $P$ of condensation points of an uncountable set in $\mathbb R^k$ is closed, and now need to show that every point of $P$ is also a limit point.

A point $p$ in a metric space $X$ is siad to be a condensation point of a subset $E$ of $X$ iff every neighborhood of $p$ contains uncountably many points of $E$.

Where can I find the complete and accurate solutions manual for Rudin, and where can I find video lectures of a comprehensive course based on this book?

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Consider a point $p$ in $P$.

Take a sequence of balls, $B_n$ of radius $\frac{1}{n}$ centered at $p$.

Because $p$ is a condensation point, for every $B_n$ there is a point $b_n$ in $P$, that is not $p$.

$b_n \to p$.

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If $p$ is a limit point of $P$ and $B$ is a neighborhood of $P$, there is an $x \in B \cap P$. Observe that $B$ is a neighborhood of $p$ and $x$. Since $x$ is a condensation point of $E, \ B\cap E$ is uncountable.
Therefore $p$ is a condensation point of $E$.

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