Definition:Continuous Mapping (Topology)
This page is about Continuous Mapping in the context of topology. For other uses, see Continuous Mapping.
Definition
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 $S_2$.
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 open set $U_2$ of $T_2$ such that $\map f x \in U_2$, there exists an open set $U_1$ of $T_1$ such that $x \in U_1$ and $f \sqbrk {U_1} \subseteq U_2$.
Definition using Neighborhoods
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 $\map f x$ in $T_2$, there exists a neighborhood $M$ of $x$ in $T_1$ such that $f \sqbrk M \subseteq N$.
Definition using Neighborhood Inverse
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 $\map f x$ in $T_2$, $f^{-1} \sqbrk N$ is a neighborhood of $x$.
Definition using Filters
The mapping $f$ is continuous at (the point) $x$ if and only if:
- for any filter $\FF$ on $T_1$ that converges to $x$, the corresponding image filter $f \sqbrk \FF$ converges to $\map f x$.
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:
- $U \in \tau_2 \implies f^{-1} \sqbrk U \in \tau_1$
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
Sources
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): continuous map
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): topological space
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): topological space