Definition:Continuous Mapping (Topology)/Everywhere/Open Sets

Definition
Let $T_1 = \left({A_1, \tau_1}\right)$ and $T_2 = \left({A_2, \tau_2}\right)$ be topological spaces.

Let $f : A_1 \to A_2$ be a mapping from $A_1$ to $A_2$.

$f$ is continuous on $A_1$ iff:
 * $U \in \tau_2 \implies f^{-1} \left({U}\right) \in \tau_1$

where $f^{-1} \left({U}\right)$ denotes the preimage of $U$ under $f$.

Also known as
If it is necessary to distinguish between multiple topologies on the same set, then the terminology $\left({\tau_1, \tau_2}\right)$-continuous can be used for the above.

Also see

 * Continuous Mapping by Open Sets for a demonstration of the equivalence of this definition and that in terms of continuity at every point.