# Axiom:Neighborhood Space Axioms

## Axioms

A neighborhood space is a set $S$ such that, for each $x \in S$, there exists a set of subsets $\NN_x$ of $S$ satisfying the following conditions:

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

These stipulations are called the neighborhood space axioms.

Each element of $\NN_x$ is called a neighborhood of $x$.