Definition:Neighborhood Space

Definition
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:

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 $\struct {S, \NN}$.

Also see

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