Principles of Mathematical Analysis, Rudin, 3th ed, Chapter 2, Problem 8

Problem

Is every point of every open set $E \subset \mathbb{R}^2$ a limit point of $E$? Answer the same question for closed sets in $\mathbb{R}^2$.

Answer

(1) Consider an open set $E \subset \mathbb{R}^2$. For $x \in E$, there exists $\epsilon>0$ such that $N_\epsilon(x) \subset E$. For any arbitrary $\delta >0$, $y = x + (0,\min(\delta/2,\epsilon/2))$ is contained in $N_\delta(x)$ and $y \neq x$, which means $x$ is a limit point of $E$.

(2) Consider a closed set $E = \{(1,1)\} \subset \mathbb{R}^2$. $(1,1)$ is not a limit point of $E$.

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