Neighborhood Space is Topological Space

Theorem
Let $\left({S, \mathcal N}\right)$ be a neighborhood space.

Let $\tau = \left\{{N: N \in \mathcal N}\right\}$ be the set of all open sets of $\left({S, \mathcal N}\right)$.

Then $\left({S, \tau}\right)$ forms a topological space.

Proof
Each of the open set axioms is examined in turn:

$O1$: Union of Open Sets
From Union of Open Sets of Neighborhood Space is Open, it follows that open set axiom $O1$ is fulfilled.

$O2$: Intersection of Open Sets
From Intersection of two Open Sets of Neighborhood Space is Open, it follows that open set axiom $O2$ is fulfilled.

$O3$: Underlying Set
From Whole Space is Open in Neighborhood Space, it follows that open set axiom $O3$ is fulfilled.

All the open set axioms are fulfilled, and the result follows.