# Definition:Continuous Mapping (Topology)/Set

## Definition

Let $T_1 = \struct {S_1, \tau_1}$ and $T_2 = \struct {S_2, \tau_2}$ be topological spaces.

Let $f: S_1 \to S_2$ be a mapping from $S_1$ to $S_2$.

Let $S$ be a subset of $S_1$.

The mapping $f$ is continuous on $S$ if and only if $f$ is continuous at every point $x \in S$.

## Also known as

If it is necessary to distinguish between multiple topologies on the same set, then the terminology $\tuple {\tau_1, \tau_2}$-continuous can be used to define a continuous mapping.

## Also see

• Results about continuous mappings can be found here.