Definition:Continuous Mapping (Topology)/Everywhere

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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 $S_2$.

Definition by Pointwise Continuity

The mapping $f$ is continuous everywhere (or simply continuous) if and only if $f$ is continuous at every point $x \in S_1$.

Definition by Open Sets

The mapping $f$ is continuous on $S_1$ if and only if:

$U \in \tau_2 \implies f^{-1} \sqbrk U \in \tau_1$

where $f^{-1} \sqbrk U$ denotes the preimage of $U$ under $f$.

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 in the context of topology can be found here.