Definition:Continuous Mapping (Metric Space)/Space/Definition 2

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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