Finite accumulation points

Question

Let $A \subset \mathbb{C}$, where $\mathbb{C}$ denotes the set of complex numbers.

Can acc($A)$ be a finite set?

(A point $x$ is an accumulation point of $A$ if every neighborhood of $x$ contains infinitely many points of $A$.)

Answer 1

Yes, it can be. For a finite $A\subset\Bbb C$, $A'=\text{acc}(A)$ is certainly finite (as an empty set). For an infinite $A$, consider $A=\{\frac 1 n:n\in\Bbb N\}$. Since $\frac 1 n\to 0$ as $n\to\infty$, $0$ is the only accumulation point of $A$. So $\text{acc}(A)=\{0\}$ is a finite set.