Topology Defined by Neighborhood System

Theorem
Let $S$ be a set.

Let $\left({\mathcal N_x}\right)_{x \mathop \in S}$ a indexed family where $\mathcal N_x$ is non-empty set of subsets of $S$.

Assume that
 * $(N1): \quad \forall x \in S, U \in \mathcal N_x: x \in U$
 * $(N2): \quad \forall x \in S, U \in \mathcal N_x, y \in U:\exists V \in \mathcal N_y: V \subseteq U$
 * $(N3): \quad \forall x \in S, U_1, U_2 \in \mathcal N_x: \exists U \in \mathcal N_x: U \subseteq U_1 \cap U_2$

Then $T = \left({S, \tau}\right)$ is a topological space where
 * $\tau = \displaystyle \left\{{\bigcup \mathcal G: \mathcal G \subseteq \bigcup_{x \mathop \in S} \mathcal N_x}\right\}$

Moreover, $\left({\mathcal N_x}\right)_{x \mathop \in S}$ is a neighborhood system of $T$.