In "Principle of Mathematical Analysis", Chapter 2, Exercise 8, Rudin stated the real-field version of the problem.
I was wondering how to prove the analogous version for the finite complex plane. In "Functions of one complex variable", Conway stated the following two standard definitions.
- For a metric space $(X,d)$ a set $G \subset X$ is open if for each $x \in G$, there exists $\epsilon > 0$ such that $B(x; \epsilon) \subset G$.
- If $A \subset X$, then a point $x \in X$ is a limit point of $A$ if there is a sequence $\{ x_n \}$ of distinct points in $A$ such that $x=\lim_{n \rightarrow \infty} x_n$.
Also, how about for accumulation points though some authors use the terms accumulation and limit points interchangeably. According to Brown and Churchill: A point $z_0$ is said to be an "accumulation point" of a set $S$ if each deleted neighborhood of $z_0$ contains at least one point of $S$.