This is from Rudin's Principal of Mathematical Analysis, Chapter 2, Problem 27.
Let $E \subseteq \mathbb{R}^k$. Let $P$ be the set of all condensation points of $E$. Let $\{ V_n \}$ be a countable base of $\mathbb{R}^k$. Let $W$ be the union of those $V_n$ for which $E \cap V_n$ is at most countable. Prove $P = W^c$ and $P$ is perfect.
I've done the part $P = W^c$ and therefore showed it is closed. But then I'm stuck at showing that every point of this set is a limit point of it.
Could you give me some vital hint for this?
Best Answer
You’ve done all the hard work. Suppose that $p\in P$ is not a limit point of $P$; then $p$ has an open nbhd $V$ such that $V\cap P=\{p\}$. Is that possible, given your construction of $P$?