Equivalence of Definitions of Topology Generated by Synthetic Basis/Definition 1 iff Definition 3

Theorem
Let $S$ be a set.

Let $\BB$ be a synthetic basis on $S$.

Let $\tau$ be the topology on $S$ generated by the synthetic basis $\BB$:
 * $\ds \tau = \set {\bigcup \AA: \AA \subseteq \BB}$

Then:
 * $\forall U \subseteq S: U \in \tau \iff \forall x \in U: \exists B \in \BB: x \in B \subseteq U$

Proof
From Set is Subset of Union: General Result, the forward implication directly follows.

We now show that the reverse implication holds.

By hypothesis, we have that:
 * $\ds U \subseteq \bigcup \set {B \in \BB: B \subseteq U}$

By Union is Smallest Superset: General Result:
 * $\ds \bigcup \set {B \in \BB: B \subseteq U} \subseteq U$

By definition of set equality:
 * $\ds U = \bigcup \set {B \in \BB: B \subseteq U}$

The result follows.