Continuity Test using Sub-Basis

Theorem
Let $\struct {X_1, \tau_1}$ and $\struct {X_2, \tau_2}$ be topological spaces.

Let $f: X_1 \to X_2$ be a mapping.

Let $\SS$ be an analytic sub-basis for $\tau_2$.

Suppose that:
 * $\forall S \in \SS: f^{-1} \sqbrk S \in \tau_1$

where $f^{-1} \sqbrk S$ denotes the preimage of $S$ under $f$.

Then $f$ is continuous.

Also see

 * Analytic Basis is Analytic Sub-Basis, of which Continuity Test using Basis is seen as a special case.