Neighborhood Space is Topological Space

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

Let $\tau = \set {N: N \in \NN}$ be the set of all open sets of $\struct {S, \NN}$.

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

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

From Union of Open Sets of Neighborhood Space is Open, it follows that {{Open-set-axiom|1} is fulfilled.

From Intersection of two Open Sets of Neighborhood Space is Open, it follows that is fulfilled.

From Whole Space is Open in Neighborhood Space, it follows that is fulfilled.

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