# Meaning when a set has an accumulation point and its negation

Let $$D'(a,r):=D(a,r)\setminus \{a\}$$ be the punctured disc with centre $$a\in \mathbb{C}$$ and radius $$r>0$$.

Let $$A\subseteq \mathbb{C}$$ be given. A point $$a\in\mathbb{C}$$ is called an accumulation point of $$A$$ if $$\forall r>0:D'(a,r)\cap A\neq \emptyset.$$

Question 1: To say that $$A$$ has an accumulation point in $$\mathbb{C}$$, does it mean that, there exists $$a\in \mathbb{C}$$ such that $$D'(a,r)\cap A\neq \emptyset$$ for all $$r>0$$?

Question 2: What does it mean to say that $$A$$ has no accumulation point in $$\mathbb{C}$$? Does it mean that, for every $$a\in \mathbb{C}$$, there exists $$r>0$$ (depending on $$a$$), such that $$D'(a,r)\cap A= \emptyset$$? Or, should the last one be $$D(a,r)\cap A= \emptyset$$?