Continuous Mappings Between Metric Spaces

Types of maps between metric spaces

Suppose $(M_{1},d_{1})$ and $(M_{2},d_{2})$ are two metric spaces.

Continuous maps

The map $f\,\colon M_{1}\to M_{2}$ is continuous if it has one (and therefore all) of the following equivalent properties:

General topological continuity

for every open set $U$ in $M_{2}$, the preimage $f^{-1}[U]$ is open in $M_{1}$

This is the general definition of continuity in topology.

Sequential continuity

if $(x_{n})$ is a sequence in $M_{1}$ that converges to $x$, then the sequence $(f(x_{n}))$ converges to $f(x)$ in $M_{2}$.

This is sequential continuity, due to Eduard Heine.

ε-δ definition

for every $x\in M_{1}$ and every $\varepsilon >0$ there exists $\delta >0$ such that for all $y$ in $M_{1}$ we have

$d_{1}(x,y)<\delta \implies d_{2}(f(x),f(y))<\varepsilon .$

This uses the (ε, δ)-definition of limit, and is due to Augustin Louis Cauchy.

Moreover, $f$ is continuous if and only if it is continuous on every compact subset of $M_{1}$.

The image of every compact set under a continuous function is compact, and the image of every connected set under a continuous function is connected.

Uniformly continuous maps

The map $f\,\colon M_{1}\to M_{2}$ is uniformly continuous if for every $\varepsilon >0$ there exists $\delta >0$ such that

$d_{1}(x,y)<\delta \implies d_{2}(f(x),f(y))<\varepsilon \quad {\mbox{for all}}\quad x,y\in M_{1}.$

Every uniformly continuous map $f\,\colon M_{1}\to M_{2}$ is continuous. The converse is true if $M_{1}$ is compact (Heine–Cantor theorem).

Uniformly continuous maps turn Cauchy sequences in $M_{1}$ into Cauchy sequences in $M_{2}$. For continuous maps this is generally wrong; for example, a continuous map from the open interval $(0,1)$ onto the real line turns some Cauchy sequences into unbounded sequences.

Lipschitz-continuous maps and contractions

Given a real number $K>0$, the map $f\,\colon M_{1}\to M_{2}$ is K-Lipschitz continuous if

$d_{2}(f(x),f(y))\leq Kd_{1}(x,y)\quad {\mbox{for all}}\quad x,y\in M_{1}.$

Every Lipschitz-continuous map is uniformly continuous, but the converse is not true in general.

If $K<1$, then $f$ is called a contraction. Suppose $M_{2}=M_{1}$ and $M_{1}$ is complete. If $f$ is a contraction, then $f$ admits a unique fixed point (Banach fixed-point theorem). If $M_{1}$ is compact, the condition can be weakened a bit: $f$ admits a unique fixed point if

$d(f(x),f(y))<d(x,y)\quad {\mbox{for all}}\quad x\neq y\in M_{1}$.

Isometries

The map $f\,\colon M_{1}\to M_{2}$ is an isometry if

$d_{2}(f(x),f(y))=d_{1}(x,y)\quad {\mbox{for all}}\quad x,y\in M_{1}$

Isometries are always injective; the image of a compact or complete set under an isometry is compact or complete, respectively. However, if the isometry is not surjective, then the image of a closed (or open) set need not be closed (or open).

Quasi-isometries

The map $f\,\colon M_{1}\to M_{2}$ is a quasi-isometry if there exist constants $A\geq 1$ and $B\geq 0$ such that

${\frac {1}{A}}d_{2}(f(x),f(y))-B\leq d_{1}(x,y)\leq Ad_{2}(f(x),f(y))+B\quad {\text{ for all }}\quad x,y\in M_{1}$

and a constant $C\geq 0$ such that every point in $M_{2}$ has a distance at most $C$ from some point in the image $f(M_{1})$.

Note that a quasi-isometry is not required to be continuous. Quasi-isometries compare the "large-scale structure" of metric spaces; they find use in geometric group theory in relation to the word metric.

Licensing

Content obtained and/or adapted from:

• Metric Space, Wikipedia under a CC BY-SA license