# 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:

 $(\text N 1)$ $:$ There exists at least one element in $\NN_x$ $\displaystyle \forall x \in S:$ $\displaystyle \NN_x \ne \O$ $(\text N 2)$ $:$ Each element of $\NN_x$ contains $x$ $\displaystyle \forall x \in S:$ $\displaystyle \forall N \in \NN_x: x \in N$ $(\text N 3)$ $:$ Each superset of $N \in \NN_x$ is also in $\NN_x$ $\displaystyle \forall x \in S: \forall N \in \NN_x:$ $\displaystyle N' \supseteq N \implies N' \in \NN_x$ $(\text N 4)$ $:$ The intersection of $2$ elements of $\NN_x$ is also in $\NN_x$ $\displaystyle \forall x \in S: \forall M, N \in \NN_x:$ $\displaystyle M \cap N \in N_x$ $(\text N 5)$ $:$ There exists $N' \subseteq N \in \NN_x$ which is $\NN_y$ of each $y \in N'$ $\displaystyle \forall x \in S: \forall N \in \NN_x:$ $\displaystyle \exists N' \in \NN_x, N' \subseteq N: \forall y \in N': N' \in \NN_y$

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

• Results about neighborhood spaces can be found here.