Definition:Continuous Mapping (Topology)/Everywhere
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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 $S_2$.
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 $\left({\tau_1, \tau_2}\right)$-continuous can be used for the above.
Also see
- Results about continuous mappings can be found here.