Equivalence of Definitions of Topology Generated by Synthetic Basis/Definition 1 iff Definition 2

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:

$\ds \forall U \subseteq S: U \in \tau \iff U = \bigcup \set {B \in \BB: B \subseteq U}$

Proof

Trivially, the reverse implication holds, as $\set {B \in \BB: B \subseteq U} \subseteq \BB$.

We now show that the forward implication holds.

Suppose $U \in \tau$.

Then, by definition:

$\ds \exists \AA \subseteq \BB: U = \bigcup \AA$

By Union is Smallest Superset: General Result:

$\forall B \in \AA: B \subseteq U$

By definition of subset, it follows that:

$\AA \subseteq \set {B \in \BB: B \subseteq U}$

From Union of Subset of Family is Subset of Union of Family:

$\ds U = \bigcup \AA \subseteq \bigcup \set {B \in \BB: B \subseteq U}$

By Union is Smallest Superset: General Result:

$\ds \bigcup \set {B \in \BB: B \subseteq U} \subseteq U$

By definition of set equality:

$\ds U = \bigcup \set {B \in \BB: B \subseteq U}$

$\blacksquare$