Prove the set of limit points in a $T_1$ space is closed.
Definitions:
Limit Point: $x$ is a limit point of a set if every open set that contains $x$ contains at least one other point of the set that is not $x$.
$T_1$-space: All pairs of disjoint points have neighborhoods not containing each others' points.
I wrote that all points in the set of all limit points are disjoint from the complement of the set of all limit points, so all the points in the complement of the set of all limit points have neighborhoods that are disjoint from all the neighborhoods of all the points in the set of all limit points, but this was marked incorrect. How should I go about proving this correctly?
