Definition:Neighborhood (Topology)/Point

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

Let $N_z$ be a subset of $S$ which contains (as a subset) an open set of $T$ which itself contains (as an element) $z$.

Then $N_z$ is a neighborhood of $z$.

That is:
 * $\exists U \in \tau: z \in U \subseteq N_z \subseteq S$

Linguistic Note
The UK English spelling of this is neighbourhood.