Definition:Continuous Mapping (Topology)/Everywhere/Open Sets

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Definition

Let $T_1 = \left({S_1, \tau_1}\right)$ and $T_2 = \left({S_2, \tau_2}\right)$ be topological spaces.

Let $f: S_1 \to S_2$ be a mapping from $S_1$ to $S_2$.


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

$U \in \tau_2 \implies f^{-1} \left({U}\right) \in \tau_1$

where $f^{-1} \left({U}\right)$ denotes the preimage of $U$ under $f$.


Also see

  • Results about continuous mappings can be found here.


Sources