Axiom:Neighborhood Space Axioms

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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 $x$.

Also see