Exercise 2.27

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Exercise 27: Suppose $E \subset \mathbb R^k$, $E$ uncountable, and let $P$ be the set of condensation points
    for $E$. $P$ is perfect, and at most a countable number of points of $E$ are not in $P$.

ghostofgarborg · Accepted Answer

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Let $\{ V_n \}$ be a countable base for $\mathbb R^k$. A point $x$ is a condensation point of $E$
    if and only if every $V_n$ that contains $x$ contains an uncountable number of points of $E$.
    Consequently, $P^c$ is the union of those $V_n$ for which $V_n \cap E$ is at most countable. 
    $P^c \cap E$ is then at most countable.

Exercise 2.27

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Exercise 2.27

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