Induced Neighborhood Space is Neighborhood Space

Theorem

Let $S$ be a set.

Let $\tau$ be a topology on $S$, thus forming the topological space $\struct {S, \tau}$.

Let $\struct {S, \NN}$ be the neighborhood space induced by $\struct {S, \tau}$.

Then $\struct {S, \NN}$ is a neighborhood space.

Proof

Let $x \in S$.

Let $\NN_x$ be the neighborhood filter of $x$.

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

Hence the result.

$\blacksquare$