Closedness v.s. convergent sequence

Question

When are the following statements equivalent?

1. A subset $A \subset X$ is closed for the given topology on $X$.

2. Every sequence $\{a_n\}$ in $A$ that converge to $a \in X$, we must have $a\in A$.

Answer 1

These are exactly equivalent in so-called sequential spaces. These include all spaces that are first countable (but also more general non first-countable spaces, e.g. all quotient images of metrisable spaces, like $\Bbb R{/}\Bbb Z$).