The textbook I am following for topology makes the following claim: "The condition that finite point sets be closed has been given a name of its own: it is called the $T_1$ axiom."
However, most explanations of the $T_1$ axiom I find online describe it as "For any two points $x,y \in X$, there exist open sets $U$ and $V$ such that $x \in U$, $x \notin V$, $y \in V$, $y \notin U$.
I don't understand how these two statements are equivalent.
Call $U_x$ the open set containing $x$ but not $y$. Take the union of the $U_x$'s to get that the complement of each point $\{y\}$ is open. Thus the point is closed.
The converse follows because when the point is closed, its complement is open.