Definition:Limit Point/Topology/Point

Definition
Let $T = \left({S, \vartheta}\right)$ be a topological space. Let $a \in S$.

A point $x \in S, x \ne a$ is called a limit point of $a$ if every open set $U \in \vartheta$ such that $x \in U$ contains $a$.

It can be seen that this is the same definition as for the definition of a limit point of a set, by requiring that the limit point for a point $a$ is defined as the limit point of the set $\left\{{a}\right\}$.