Topological Space Induced by Neighborhood Space is Topological Space

Theorem

Let $\struct {S, \NN}$ be a neighborhood space.

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

Then $\struct {S, \tau}$ is a topological space.

Proof

From Neighborhood Space is Topological Space, $\struct {S, \NN}$ is a topological space.

Consequently, the open sets of $\struct {S, \tau}$ are exactly the open sets of $\struct {S, \NN}$.

$\blacksquare$