Synthetic Sub-Basis and Analytic Sub-Basis are Compatible

Theorem

Let $\struct {X, \tau}$ be a topological space.

Let $\SS \subseteq \powerset X$, where $\powerset X$ denotes the power set of $X$.

Then $\SS$ is an analytic sub-basis for $\tau$ if and only if $\tau$ is the topology on $X$ generated by the synthetic sub-basis $\SS$.

Proof

Necessary Condition

Follows directly from the definitions of the generated topology and an analytic sub-basis.

$\Box$

Sufficient Condition

Suppose that $\SS$ is an analytic sub-basis for $\tau$.

Let $\tau'$ be the topology on $X$ generated by the synthetic sub-basis $\SS$.

By definition $1$ of the generated topology and the definition of an analytic sub-basis, we have $\tau \subseteq \tau'$.

By definition $2$ of the generated topology, it follows that $\tau' \subseteq \tau$.

By definition of set equality:

$\tau = \tau'$

$\blacksquare$