Sequential characterization of closed sets

Sequential characterization of closed sets: Let $(X,d)$ be a metric space and let $A\subseteq X$. Then $A$ is closed if and only if for every convergent sequence $(a_n)$ in $A$ with $a_n\to x$ in $X$, one has $x\in A$.

This is the standard “sequentially closed equals closed” principle for metric spaces, and it pairs naturally with the sequential characterization of closure .