What is the intuitive sense of a closed set on a metric space?

Question

A subset $A$ of $X$ is closed in $(X, d)$ if and only if every convergent sequence of points in $A$ converges to a point in $A$. How is this definition consistent with the open ball definition of closed/openness? As far I know a subset $A$ in $X$ is closed, if it's complement is open, and a subset is open if $\exists \epsilon > 0$ s.t every element in an open ball of radius $\epsilon$ is completely in that set $\forall x \in subset $. What is the intuition that connects both these definitions?

Answer 1

If a set $A$ is not closed, then $A^\complement$ is not open, and therefore there is some $p\in A^\complement$ such that no open ball centered at $P$ is contained in $A^\complement$. But this is the same thing as asserting that every open ball centered at $p$ contains points of $A$. From this, it is not hard to prove that there is a sequence of points of $A$ which converges to $p$… which does not belong to $A$.