Continuous Mapping is Continuous on Induced Topological Spaces
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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 $3.1.11$