\begin{definition}[Definition:Neighborhood Space]
Let $S$ be a set.
For each $x \in S$, let there be given a set $\NN_x$ of subsets of $S$ which satisfy the neighborhood space axioms:
{{:Axiom:Neighborhood Space Axioms}}
The sets $\NN_x$ are the neighborhoods of $x$ in $S$.
Let $\NN$ be the set of open sets of $S$:
:$\NN = \leftset {U \subseteq S: U}$ is a neighborhood of each of its elements$\rightset {}$
The set $S$ together with $\NN$ is called a '''neighborhood space''' and is denoted $\struct {S, \NN}$.
\end{definition}