I keep seeing the closure of a set to be defined as follows:
$\textbf{Definition:}$ Let $(X, d)$ be a metric space. Also, let $A\subseteq X$.
Then, $x\in cl(A)$ iff $\forall \epsilon>0$, $B(x;\epsilon)\cap A\neq \phi$ $\tag1$
I personally use the definition at the top shown below which I tagged as $(2)$.
$\textbf{Question:}$ How are these definitions equivalent (i.e. (1) and (2))? Any help would be appreciated!
\begin{align} x\in cl(A) \leftrightarrow \forall \text{ neighborhood } N \text{ of } x, N\cap A\neq \phi \tag2 \end{align}
Also, I use the following definition of a neighborhood: $N$ is a $\textbf{neighborhood}$ of $a$ given a metric space $(X, d)$ iff $\exists \delta\in \mathbb{R}^{>0}$ ST $B(a;\delta)\subseteq N\subseteq X$.
According to your definition of neighborhood, every open ball centered at x is a neighborhood of x and every neighborhood of x contains an open ball centered at x.
Thus if every neighborhood of x intersects A then every open ball will intersect A and if every open ball centered at x intersects A, then every neighborhood of x intersects A because it contains an open ball centered at x.
Therefore these two definitions are equivalent.