Definition:Neighborhood Space Induced by Topological Space

Theorem
Let $S$ be a set.

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

For all $x \in S$, let $\mathcal N_x$ be the neighborhood filter of $x$ in $S$.

Let $\mathcal N$ be the set of subsets of $S$ such that:
 * $\forall N \in \mathcal N: N$ is a neighborhood of each of its points

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

Also known as
Some sources refer to $\left({S, \mathcal N}\right)$ in this context as the neighborhood space given rise to by $\left({S, \tau}\right)$.

Also see

 * Induced Neighborhood Space is Neighborhood Space