Definition:Generic Point of Topological Space

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

Let $x \in S$ be an element of $S$.

Definition 1
The point $x$ is a generic point of $T$ the closure of the singleton $\left\{{x}\right\}$ is $S$.

Definition 2
The point $x$ is a generic point of $T$ $x$ is contained in every non-empty open subset of $T$.

Also see

 * Equivalence of Definitions of Generic Point of Topological Space