Definition:Continuous Mapping (Topology)

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This page is about continuous mappings in the context of the field of topology. For other uses, see Definition:Continuous Mapping.

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 $S_2$.


Continuous at a Point

Let $x \in S_1$.

Definition using Open Sets

The mapping $f$ is continuous at (the point) $x$ (with respect to the topologies $\tau_1$ and $\tau_2$) if and only if:

For every neighborhood $N$ of $f \left({x}\right)$ in $T_2$, there exists a neighborhood $M$ of $x$ in $T_1$ such that $f \left({M}\right) \subseteq N$.


Definition using Filters

The mapping $f$ is continuous at (the point) $x$ if and only if for any filter $\mathcal F$ on $T_1$ that converges to $x$, the corresponding image filter $f \left({\mathcal F}\right)$ converges to $f \left({x}\right)$.


If necessary, we can say that $f$ is $\left({\tau_1, \tau_2}\right)$-continuous at $x$.


Continuous on a Set

Let $S$ be a subset of $S_1$.

The mapping $f$ is continuous on $S$ if and only if $f$ is continuous at every point $x \in S$.


Continuous Everywhere

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:

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

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


Also see

  • Results about continuous mappings can be found here.