Definition:Neighborhood (Topology)/Set

Definition
Let $\left({X, \vartheta}\right)$ be a topological space. Let $A \subseteq X$ be a subset of $X$.

A neighborhood of $A$, which can be denoted $N_A$, is any subset of $X$ containing an open set which itself contains $A$.

That is:
 * $\exists U \in \vartheta: A \subseteq U \subseteq N_A \subseteq X$

Linguistic Note
The UK English spelling of this is neighbourhood.