# 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:

 $(N1)$ $:$ There exists at least one element in $\mathcal N_x$ $\displaystyle \forall x \in S:$ $\displaystyle \mathcal N_x \ne \varnothing$ $(N2)$ $:$ Each element $x$ is in its own $\mathcal N_x$ $\displaystyle \forall x \in S:$ $\displaystyle \forall N \in \mathcal N_x: x \in N$ $(N3)$ $:$ Each superset of $N \in \mathcal N_x$ is also in $\mathcal N_x$ $\displaystyle \forall x \in S: \forall N \in \mathcal N_x:$ $\displaystyle N' \supseteq N \implies N' \in \mathcal N_x$ $(N4)$ $:$ Intersection of $2$ elements of $\mathcal N_x$ is also in $\mathcal N_x$ $\displaystyle \forall x \in S: \forall M, N \in \mathcal N_x:$ $\displaystyle M \cap N \in N_x$ $(N5)$ $:$ Exists $N' \subseteq N \in \mathcal N_x$ which is $\mathcal N_y$ of each $y \in N'$ $\displaystyle \forall x \in S: \forall N \in \mathcal N_x:$ $\displaystyle \exists N' \in \mathcal N_x, N' \subseteq N: \forall y \in N': N' \in \mathcal N_y$

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

• Results about neighborhood spaces can be found here.