Definition:Continuous Mapping (Metric Space)/Space/Definition 2
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Definition
Let $M_1 = \struct {A_1, d_1}$ and $M_2 = \struct {A_2, d_2}$ be metric spaces.
Let $f: A_1 \to A_2$ be a mapping from $A_1$ to $A_2$.
$f$ is continuous from $\struct {A_1, d_1}$ to $\struct {A_2, d_2}$ if and only if:
- for every $U \subseteq A_2$ which is open in $M_2$, $f^{-1} \sqbrk U$ is open in $M_1$.
By definition, this is equivalent to the continuity of $f$ with respect to the induced topologies on $A_1$ and $A_2$.
Also known as
A mapping which is continuous from $\struct {A_1, d_1}$ to $\struct {A_2, d_2}$ can also be referred to as $\tuple {d_1, d_2}$-continuous.
Also see
Sources
- 1967: George McCarty: Topology: An Introduction with Application to Topological Groups ... (previous) ... (next): Chapter $\text{III}$: Metric Spaces: More About Continuity
- 1975: Bert Mendelson: Introduction to Topology (3rd ed.) ... (previous) ... (next): Chapter $2$: Metric Spaces: $\S 6$: Open Sets and Closed Sets: Theorem $6.3$
- 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $2$: Continuity generalized: metric spaces: $2.3$: Open sets in metric spaces: Proposition $2.3.13$
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- 1953: Walter Rudin: Principles of Mathematical Analysis ... (previous) ... (next): $4.8$