# Questions about Theorem 2.43 in Baby Rudin

Theorem 2.43 $$\hspace{5 pt}$$ Let $$P$$ be a nonempty perfect set in $$\mathbb{R}^k$$. Then $$P$$ is uncountable.

Proof $$\hspace{5 pt}$$ Since $$P$$ has limit points, $$P$$ must be infinite. Suppose $$P$$ is countable, and denote the points of $$P$$ by $$\mathbf{x_1}, \mathbf{x_2}, \mathbf{x_3}, \ldots$$. We shall construct a sequence $$\{V_n\}$$ of neighborhoods as follows.

Let $$V_1$$ be any neighborhood of $$x_1$$.

If $$V_1$$ consists of all $$\mathbf{y} \in \mathbb{R}^k$$ such that $$|\mathbf{y} - \mathbf{x_1}| < r$$, the closure $$\overline{V_1}$$ of $$V_1$$ is the set of all $$\mathbf{y} \in \mathbb{R}^k$$ such that $$|\mathbf{y} - \mathbf{x_1}| \leq r$$. etc...

Here is the problem. I know every neighborhood is an open set from Theorem 2.19, but why the closure of $$V_1$$ is the set of all y such that $$|\mathbf{y} - \mathbf{x_1}| \leq r$$? Can someone prove it? Thanks in advance.

• What are the accumulation points of $V_1$? Jun 29, 2017 at 8:19
• Sorry, I don't know what is the accumulation point, I think there is no definition in baby rudin or I've not read there.@FrancescoPolizzi Jun 29, 2017 at 8:21
• You should study Rudin's book more carefully. Accumulation points are considered in Definition 2.18, and in the same place you can also find the definition of closed set, namely a set containing all its accumulation points. Jun 29, 2017 at 8:27
• I think you mean limit points, right? A closed set contains all its limit points. Jun 29, 2017 at 9:24
• I got it. The limit points of $V_1$ are $\mathbf{y} \in \mathbb{R}^k$ such that $|\mathbf{y} - \mathbf{x_1}| = r$ and the points such that $|\mathbf{y} - \mathbf{x_1}| < r$. Jun 29, 2017 at 9:37

Let $C$ be the set $\{y~:~\lvert y-x\rvert\leq r\}$. Every point in $C$ is a limit point of $V$, so $C\subset\bar{V}$. On the other hand, any point $z$ in $R^{k}-C$ has a neighborhood disjoint from $C$, so $z$ is not a limit point of $V$ (nor a member of $V$). But $\bar{V}=V\cup V'$, where $V'$ is the set of limit points of $V$.