Functions between metric spaces

Functions between metric spaces[edit]

Euler diagram of types of functions between metric spaces.

Unlike in the case of topological spaces or algebraic structures such as groups or rings, there is no single "right" type of structure-preserving function between metric spaces. Instead, one works with different types of functions depending on one's goals. Throughout this section, suppose that $(M_{1},d_{1})$ and $(M_{2},d_{2})$ are two metric spaces. The words "function" and "map" are used interchangeably.

Isometries[edit]

One interpretation of a "structure-preserving" map is one that fully preserves the distance function:

A function

f:M_{1}\to M_{2}

is distance-preserving^[12] if for every pair of points

x

and

y

M 1

d_{2}(f(x),f(y))=d_{1}(x,y).

It follows from the metric space axioms that a distance-preserving function is injective. A bijective distance-preserving function is called an isometry.^[13] One perhaps non-obvious example of an isometry between spaces described in this article is the map $f:(\mathbb {R} ^{2},d_{1})\to (\mathbb {R} ^{2},d_{\infty })$ defined by

f(x,y)=(x+y,x-y).

If there is an isometry between the spaces $M 1$ and $M 2$ , they are said to be isometric. Metric spaces that are isometric are essentially identical.

Continuous maps[edit]

On the other end of the spectrum, one can forget entirely about the metric structure and study continuous maps, which only preserve topological structure. There are several equivalent definitions of continuity for metric spaces. The most important are:

Topological definition. A function $f\,\colon M_{1}\to M_{2}$ is continuous if for every open set $U$ in $M 2$ , the preimage $f^{-1}(U)$ is open.
Sequential continuity. A function $f\,\colon M_{1}\to M_{2}$ is continuous if whenever a sequence $(x n)$ converges to a point $x$ in $M 1$ , the sequence $f(x_{1}),f(x_{2}),\ldots$ converges to the point $f (x)$ in $M 2$ .

(These first two definitions are not equivalent for all topological spaces.)

ε–δ definition. A function $f\,\colon M_{1}\to M_{2}$ is continuous if for every point $x$ in $M 1$ and every $ε > 0$ there exists $δ > 0$ such that for all $y$ in $M 1$ we have $d_{1}(x,y)<\delta \implies d_{2}(f(x),f(y))<\varepsilon .$

A homeomorphism is a continuous map whose inverse is also continuous; if there is a homeomorphism between $M 1$ and $M 2$ , they are said to be homeomorphic. Homeomorphic spaces are the same from the point of view of topology, but may have very different metric properties. For example, $\mathbb {R}$ is unbounded and complete, while $(0, 1)$ is bounded but not complete.

Uniformly continuous maps[edit]

A function $f\,\colon M_{1}\to M_{2}$ is uniformly continuous if for every real number $ε > 0$ there exists $δ > 0$ such that for all points $x$ and $y$ in $M 1$ such that $d(x,y)<\delta$ , we have

d_{2}(f(x),f(y))<\varepsilon .

The only difference between this definition and the ε–δ definition of continuity is the order of quantifiers: the choice of δ must depend only on ε and not on the point $x$ . However, this subtle change makes a big difference. For example, uniformly continuous maps take Cauchy sequences in $M 1$ to Cauchy sequences in $M 2$ . In other words, uniform continuity preserves some metric properties which are not purely topological.

On the other hand, the Heine–Cantor theorem states that if $M 1$ is compact, then every continuous map is uniformly continuous. In other words, uniform continuity cannot distinguish any non-topological features of compact metric spaces.

Lipschitz maps and contractions[edit]

A Lipschitz map is one that stretches distances by at most a bounded factor. Formally, given a real number $K > 0$ , the map $f\,\colon M_{1}\to M_{2}$ is $K$ -Lipschitz if

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

Lipschitz maps are particularly important in metric geometry, since they provide more flexibility than distance-preserving maps, but still make essential use of the metric.^[14] For example, a curve in a metric space is rectifiable (has finite length) if and only if it has a Lipschitz reparametrization.

A 1-Lipschitz map is sometimes called a nonexpanding or metric map. Metric maps are commonly taken to be the morphisms of the category of metric spaces.

A $K$ -Lipschitz map for $K < 1$ is called a contraction. The Banach fixed-point theorem states that if $M$ is a complete metric space, then every contraction $f:M\to M$ admits a unique fixed point. If the metric space $M$ is compact, the result holds for a slightly weaker condition on $f$ : a map $f:M\to M$ 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}.

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