Equivalence of Definitions of Topology

Definition 1 implies Definition 2
Let $\tau$ be a topology on $S$ by definition 1.

$\text O 1$ is the same as $\text O 1'$, so $\text O 1'$ holds for $\tau$.

From General Intersection Property of Topological Space:
 * if $\text O 2$ holds, then $\text O 2'$ holds.

Thus $\tau$ is a topology on $S$ by definition 2.

Definition 2 implies Definition 1
Let $\tau$ be a topology on $S$ by definition 2.

$\text O 1'$ is the same as $\text O 1$, so $\text O 1$ holds for $\tau$.

$\text O 2'$ states that the intersection of any finite subset of $\tau$ is an element of $\tau$.

This applies when the subset of $\tau$ contains exactly $2$ sets.

Thus $\text O 2$ is a direct consequence of $\text O 2'$.

Also as a consequence of $\text O 2$, it follows that the intersection of an empty subset of $\tau$ is an element of $\tau$.

From Intersection of Empty Set it follows that $S \in \tau$.

So $\text O 3$ is a direct consequence of $\text O 2'$.

Thus all the open set axioms hold.

Thus $\tau$ is a topology on $S$ by definition 1.