Continuity Test using Sub-Basis/Proof 1

Proof
Define:
 * $\ds \BB = \set {\bigcap \AA: \AA \subseteq \SS, \AA \text{ is finite} } \subseteq \powerset {X_2}$

Let $B \in \BB$.

Then there exists a finite subset $\AA \subseteq \SS$ such that:
 * $\ds B = \bigcap \AA$

Hence:

Define:
 * $\ds \tau = \set {\bigcup \AA: \AA \subseteq \BB} \subseteq \powerset {X_2}$

By the definition of an analytic sub-basis, we have $\tau_2 \subseteq \tau$.

Let $U \in \tau_2$.

Then $U \in \tau$; therefore:
 * $\ds \exists \AA \subseteq \BB: U = \bigcup \AA$

Hence:

That is, $f$ is continuous.