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 $x$.

1975: Bert Mendelson: Introduction to Topology (3rd ed.) ... (previous) ... (next): Chapter $3$: Topological Spaces: $\S 3$: Neighborhoods and Neighborhood Spaces: Definition $3.4$

\((\text N 1)\)	$:$	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 $y \in N'$	\(\ds \forall x \in S: \forall N \in \NN_x:\)	\(\ds \exists N' \in \NN_x, N' \subseteq N: \forall y \in N': N' \in \NN_y \)