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 $D(a,r)\cap A= \emptyset$?


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