Induced Neighborhood Space is Neighborhood Space

Theorem
Let $S$ be a set.

Let $\tau$ be a topology on $S$, thus forming the topological space $\left({S, \tau}\right)$.

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

Then $\left({S, \mathcal N}\right)$ is a neighborhood space.

Proof
Let $x \in S$.

Let $\mathcal N_x$ be the neighborhood filter of $x$.

From Basic Properties of Neighborhood in Topological Space, $\mathcal N_x$ fulfils the neighborhood space axioms.

Hence the result.