# 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 $\mathcal N_x$ of $S$ satisfying the following conditions:

 $(N1)$ $:$ 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 $y \in N'$ $\displaystyle \forall x \in S: \forall N \in \mathcal N_x:$ $\displaystyle \exists N' \in \mathcal N_x, N' \subseteq N: \forall y \in N': N' \in \mathcal N_y$

These stipulations are called the neighborhood space Axioms.

Each element of $\mathcal N_x$ is called a neighborhood of $x$.