Metric space

In mathematics, a metric space is a set together with a notion of distance between its elements, usually called points. The distance is measured by a function called a metric or distance function.^[1] Metric spaces are the most general setting for studying many of the concepts of mathematical analysis and geometry.

The most familiar example of a metric space is 3-dimensional Euclidean space with its usual notion of distance. Other well-known examples are a sphere equipped with the angular distance and the hyperbolic plane. A metric may correspond to a metaphorical, rather than physical, notion of distance: for example, the set of 100-character Unicode strings can be equipped with the Hamming distance, which measures the number of characters that need to be changed to get from one string to another.

Since they are very general, metric spaces are a tool used in many different branches of mathematics. Many types of mathematical objects have a natural notion of distance and therefore admit the structure of a metric space, including Riemannian manifolds, normed vector spaces, and graphs. In abstract algebra, the p-adic numbers arise as elements of the completion of a metric structure on the rational numbers. Metric spaces are also studied in their own right in metric geometry^[2] and analysis on metric spaces.^[3]

Many of the basic notions of mathematical analysis, including balls, completeness, as well as uniform, Lipschitz, and Hölder continuity, can be defined in the setting of metric spaces. Other notions, such as continuity, compactness, and open and closed sets, can be defined for metric spaces, but also in the even more general setting of topological spaces.

Topological definition. A function $f\,\colon M_{1}\to M_{2}$ is continuous if for every open set $U$ in $M 2$ , the $f^{-1}(U)$ is open.

preimage

. 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 $M 2$ .

Sequential continuity

translation invariant: $d(x,y)=d(x+a,y+a)$ for every $x$ , $y$ , and $a$ in $X$ ; and

: $d(\alpha x,\alpha y)=|\alpha |d(x,y)$ for every $x$ and $y$ in $X$ and real number $α$ ;

absolutely homogeneous

Further examples and applications[edit]

Graphs and finite metric spaces[edit]

A metric space is discrete if its induced topology is the discrete topology. Although many concepts, such as completeness and compactness, are not interesting for such spaces, they are nevertheless an object of study in several branches of mathematics. In particular, finite metric spaces (those having a finite number of points) are studied in combinatorics and theoretical computer science.^[23] Embeddings in other metric spaces are particularly well-studied. For example, not every finite metric space can be isometrically embedded in a Euclidean space or in Hilbert space. On the other hand, in the worst case the required distortion (bilipschitz constant) is only logarithmic in the number of points.^[24]^[25]

For any undirected connected graph $G$ , the set $V$ of vertices of $G$ can be turned into a metric space by defining the distance between vertices $x$ and $y$ to be the length of the shortest edge path connecting them. This is also called shortest-path distance or geodesic distance. In geometric group theory this construction is applied to the Cayley graph of a (typically infinite) finitely-generated group, yielding the word metric. Up to a bilipschitz homeomorphism, the word metric depends only on the group and not on the chosen finite generating set.^[15]

Distances between mathematical objects[edit]

In modern mathematics, one often studies spaces whose points are themselves mathematical objects. A distance function on such a space generally aims to measure the dissimilarity between two objects. Here are some examples: