Let $A$ be a subset, $A \subset \mathbb{R}$. A point $a \in \mathbb{\overline{R}}$ is a limit point(or accumulation point) of $A$ if every neighbourhood of $a$ contains at least one point of $A$ different from $a$ itself
I cannot unerstand this definition very well. For this I will draw a picture. I have a set $A$, and two neighbourhoods $V$ and $W$.
case I. For the neighbourhood $V$ our definition is verified because $V \cap A \neq \emptyset$
case II. neighbourhood $W$ is not ok because $A \cap W =\emptyset$.
Why in the definition is specified the word every? I can find at least a neighbourhood $U$ for that $U \cap A =\emptyset$.
Thanks :)