Continuous Mapping is Continuous on Induced Topological Spaces

Theorem

Let $M_1 = \struct {A_1, d_1}$ and $M_2 = \struct {A_2, d_2}$ be metric spaces.

Let $\tau_{d_1}$ and $\tau_{d_2}$ be the topologies induced by the metrics $d_1$ and $d_2$.

Let $T_1 = \struct {A_1, \tau_{d_1} }$ and $T_2 = \struct {A_2, \tau_{d_2} }$ be the resulting topological spaces.

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

Then $f$ is $\tuple {d_1, d_2}$-continuous if and only if $f$ is $\tuple {\tau_{d_1}, \tau_{d_2} }$-continuous.

Proof

Follows directly from:

the open set definition of continuity on a metric space

the definition of continuity on a topological space.

$\blacksquare$

Sources

• 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $3$: Continuity generalized: topological spaces: $3.1$: Topological Spaces: Proposition $3.1.11$