Union from Synthetic Basis is Topology/Proof 1

Proof
We proceed to verify the open set axioms for $\tau$ to be a topology on $S$.

$(O1):$ Union of Open Sets
Let $\AA \subseteq \tau$.

It is to be shown that:
 * $\displaystyle \bigcup \AA \in \tau$

Define:
 * $\displaystyle \AA' = \bigcup_{U \mathop \in \AA} \set {B \in \BB: B \subseteq U}$

By Union is Smallest Superset: Family of Sets, it follows that $\AA' \subseteq \BB$.

Hence, by Equivalence of Definitions of Topology Generated by Synthetic Basis and General Distributivity of Set Union:
 * $\displaystyle \bigcup \AA = \bigcup_{U \mathop \in \AA} \bigcup \set {B \in \BB: B \subseteq U} = \bigcup \AA' \in \tau$

$(O2):$ Pairwise Intersection of Open Sets
Let $U, V \in \tau$.

It is to be shown that:
 * $U \cap V \in \tau$

Define:
 * $\OO = \set {A \cap B: A, B \in \BB, \, A \subseteq U, \, B \subseteq V}$

By the definition of a synthetic basis:
 * $\forall A, B \in \BB: A \cap B \in \tau$

Hence, by the definition of a subset, it follows that $\OO \subseteq \tau$.

By open set axiom $(1)$, which has been verified above for $\tau$, we have:
 * $\displaystyle \bigcup \OO \in \tau$

Since set intersection preserves subsets, we have:
 * $\displaystyle \forall W \in \OO: \exists A, B \in \BB: W = A \cap B \subseteq U \cap V$

By Union is Smallest Superset: General Result, it follows that:
 * $\displaystyle \bigcup \OO \subseteq U \cap V$

By the definition of $\tau$, it follows from Set is Subset of Union: General Result that:
 * $\displaystyle \forall x \in U \cap V: \exists A, B \in \BB: A \subseteq U, \, B \subseteq V: x \in A \cap B \subseteq \bigcup \OO$

That is, by the definition of a subset:
 * $\displaystyle U \cap V \subseteq \bigcup \mathcal O$

By definition of set equality:
 * $\displaystyle U \cap V = \bigcup \OO \in \tau$

$(O3):$ Set Itself
By the definition of a synthetic basis, $\mathcal B$ is a cover for $S$.

By Equivalent Conditions for Cover by Collection of Subsets, it follows that:
 * $\displaystyle S = \bigcup \BB \in \tau$

Also see

 * Definition:Topology Generated by Synthetic Basis
 * Definition:Topology Generated by Synthetic Sub-Basis