Equivalence of Definitions of Topology Generated by Synthetic Basis/Definition 1 iff Definition 3
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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.
$\blacksquare$