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

Theorem
Let $S$ be a set.

Let $\mathcal B$ be a synthetic basis on $S$.

Let $\tau$ be the topology on $S$ generated by the synthetic basis $\mathcal B$:
 * $\displaystyle \tau = \left\{{\bigcup \mathcal A: \mathcal A \subseteq \mathcal B}\right\}$

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

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:
 * $\displaystyle U \subseteq \bigcup \left\{{B \in \mathcal B: B \subseteq U}\right\}$

By Union is Smallest Superset: General Result:
 * $\displaystyle \bigcup \left\{{B \in \mathcal B: B \subseteq U}\right\} \subseteq U$

By definition of set equality:
 * $\displaystyle U = \bigcup \left\{{B \in \mathcal B: B \subseteq U}\right\}$

The result follows.