Definition:Neighborhood Space

Definition
Let $S$ be a set.

For each $x \in S$, let there be given a set $\mathcal N_x$ of subsets of $S$ which satisfy the neighborhood space axioms:

The sets $\mathcal N_x$ are the neighborhoods of $x$ in $S$.

Let $\mathcal N$ be the set of open sets of $S$:
 * $\mathcal N = \left\{{U \subseteq S: U}\right.$ is a neighborhood of each of its elements $\left.{}\right\}$

The set $S$ together with $\mathcal N$ is called a neighborhood space and is denoted $\left({S, \mathcal N}\right)$.

Also see

 * Basic Properties of Neighborhood in Topological Space
 * Definition:Neighborhood (Neighborhood Space)