Topological Space Induced by Neighborhood Space is Topological Space

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

Let $\left({S, \tau}\right)$ be the topological space induced by $\left({S, \mathcal N}\right)$.

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

Proof
From Neighborhood Space is Topological Space, $\left({S, \mathcal N}\right)$ is a topological space.

Consequently, the open sets of $\left({S, \tau}\right)$ are exactly the open sets of $\left({S, \mathcal N}\right)$.