Metric space

In mathematics, a metric space is a set together with a notion of distance between its points. The distance is measured by a function called a metric or distance function. Metric spaces are a 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.

Metric spaces appear in many different branches of mathematics. For example, Riemannian manifolds, normed vector spaces, and graphs may be viewed as metric spaces. In abstract algebra, the field of p-adic numbers is the completion of the field of rational numbers with respect to a certain metric. Metric spaces are also studied in their own right in metric geometry and analysis on metric spaces.

Many notions of analysis, including balls, completeness, as well as uniform, Lipschitz, and Hölder continuity can be defined for 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.

To see the utility of different notions of distance, consider the surface of the Earth as a set of points. We can measure the distance between two such points by the length of the shortest path along the surface, "as the crow flies"; this is particularly useful for shipping and aviation. We can also measure the straight-line distance between two points through the Earth's interior; this notion is, for example, natural in seismology, since it roughly corresponds to the length of time it takes for seismic waves to travel between those two points.

The notion of distance encoded by the metric space axioms has relatively few requirements. This generality gives metric spaces a lot of flexibility. At the same time, the notion is strong enough to encode many intuitive facts about what distance means. This means that general results about metric spaces can be applied in many different contexts.

Like many fundamental mathematical concepts, the metric on a metric space can be interpreted in many different ways. A particular metric may not be best thought of as measuring physical distance, but, instead, as the cost of changing from one state to another (as with Wasserstein metrics on spaces of measures) or the degree of difference between two objects (for example, the Hamming distance between two strings of characters, or the Gromov-Hausdorff distance between metric spaces themselves).

Formally, a metric space is an ordered pair (M, d) where M is a set (whose elements are called points), and d is a metric on M, i.e., a function\[d\,\colon M \times M \to \mathbb{R}\]satisfying the following axioms for all points \(x,y,z \in M\):

1. The distance from a point to itself is zero: \[d(x, x) = 0\]
2. (Positivity) The distance between two distinct points is always positive: \[\text{If }x \neq y\text{, then }d(x, y)>0\]
3. (Symmetry) The distance from x to y is always the same as the distance from y to x: \[d(x, y) = d(y, x)\]
4. The triangle inequality holds: \[d(x, z) \leq d(x, y) + d(y, z)\]This is a natural property of both physical and metaphorical notions of distance: you can arrive at z from x by taking a detour through y, but this will not make your journey any shorter than the direct path.

It is traditional to write M as an abbreviation for (M, d) when it is understood what d is.

Arthur Cayley, in his article "On Distance", extended metric concepts beyond Euclidean geometry into domains bounded by a conic in a projective space. His distance was given by logarithm of a cross ratio. Any projectivity leaving the conic stable also leaves the cross ratio constant, so isometries are implicit. This method provides models for elliptic geometry and hyperbolic geometry, and Felix Klein, in several publications, established the field of non-euclidean geometry through the use of the Cayley-Klein metric.

The idea of an abstract space with metric properties was addressed in 1906 by René Maurice Fréchet and the term metric space was coined by Felix Hausdorff in 1914.

Fréchet's work laid the foundation for understanding convergence, continuity, and other key concepts in non-geometric spaces. This allowed mathematicians to study functions and sequences in a broader and more flexible way. This was important for the growing field of functional analysis. Mathematicians like Hausdorff and Stefan Banach further refined and expanded the framework of metric spaces. Hausdorff introduced topological spaces as a generalization of metric spaces. Banach's work in functional analysis heavily relied on the metric structure. Over time, metric spaces became a central part of modern mathematics. They have influenced various fields including topology, geometry, and applied mathematics. Metric spaces continue to play a crucial role in the study of abstract mathematical concepts.

A distance function is enough to define notions of closeness and convergence that were first developed in real analysis. Properties that depend on the structure of a metric space are referred to as metric properties. Every metric space is also a topological space, and some metric properties can also be rephrased without reference to distance in the language of topology; that is, they are really topological properties.

For any point x in a metric space M and any real number r > 0, the open ball of radius r around x is defined to be the set of points that are strictly less than distance r from x: \[B_r(x)=\{y \in M : d(x,y) < r\}.\] This is a natural way to define a set of points that are relatively close to x. Therefore, a set \(N \subseteq M\) is a neighborhood of x (informally, it contains all points "close enough" to x) if it contains an open ball of radius r around x for some r > 0.

An open set is a set which is a neighborhood of all its points. It follows that the open balls form a base for a topology on M. In other words, the open sets of M are exactly the unions of open balls. As in any topology, closed sets are the complements of open sets. Sets may be both open and closed as well as neither open nor closed.

This topology does not carry all the information about the metric space. For example, the distances d₁, d₂, and d_∞ defined above all induce the same topology on \(\R^2\), although they behave differently in many respects. Similarly, \(\R\) with the Euclidean metric and its subspace the interval (0, 1) with the induced metric are homeomorphic but have very different metric properties.

Conversely, not every topological space can be given a metric. Topological spaces which are compatible with a metric are called metrizable and are particularly well-behaved in many ways: in particular, they are paracompact Hausdorff spaces (hence normal) and first-countable. The Nagata-Smirnov metrization theorem gives a characterization of metrizability in terms of other topological properties, without reference to metrics.

Convergence of sequences in Euclidean space is defined as follows:

A sequence (x_n) converges to a point x if for every ε > 0 there is an integer N such that for all n > N, d(x_n, x) < ε.

Convergence of sequences in a topological space is defined as follows:

A sequence (x_n) converges to a point x if for every open set U containing x there is an integer N such that for all n > N, \(x_n \in U\).

In metric spaces, both of these definitions make sense and they are equivalent. This is a general pattern for topological properties of metric spaces: while they can be defined in a purely topological way, there is often a way that uses the metric which is easier to state or more familiar from real analysis.

Informally, a metric space is complete if it has no "missing points": every sequence that looks like it should converge to something actually converges.

To make this precise: a sequence (x_n) in a metric space M is a Cauchy sequence if for every ε > 0 there is an integer N such that for all m, n > N, d(x_m, x_n) < ε. By the triangle inequality, any convergent sequence is a Cauchy sequence: if x_m and x_n are both less than ε away from the limit, then they are less than 2ε away from each other. If the converse is true, every Cauchy sequence in M converges, then M is complete.

Euclidean spaces are complete, as is \(\R^2\) with the other metrics described above. Two examples of spaces which are not complete are (0, 1) and the rationals, each with the metric induced from \(\R\). One can think of (0, 1) as "missing" its endpoints 0 and 1. The rationals are missing all the irrationals, since any irrational has a sequence of rationals converging to it in \(\R\) (for example, its successive decimal approximations). These examples show that completeness is not a topological property, since \(\R\) is complete but the homeomorphic space (0, 1) is not.

This notion of "missing points" can be made precise. In fact, every metric space has a unique completion, which is a complete space that contains the given space as a dense subset. For example, [0, 1] is the completion of (0, 1), and the real numbers are the completion of the rationals.

Since complete spaces are generally easier to work with, completions are important throughout mathematics. For example, in abstract algebra, the p-adic numbers are defined as the completion of the rationals under a different metric. Completion is particularly common as a tool in functional analysis. Often one has a set of nice functions and a way of measuring distances between them. Taking the completion of this metric space gives a new set of functions which may be less nice, but nevertheless useful because they behave similarly to the original nice functions in important ways. For example, weak solutions to differential equations typically live in a completion (a Sobolev space) rather than the original space of nice functions for which the differential equation actually makes sense.

A metric space M is bounded if there is an r such that no pair of points in M is more than distance r apart. The least such r is called the diameter of M.

The space M is called precompact or totally bounded if for every r > 0 there is a finite cover of M by open balls of radius r. Every totally bounded space is bounded. To see this, start with a finite cover by r-balls for some arbitrary r. Since the subset of M consisting of the centers of these balls is finite, it has finite diameter, say D. By the triangle inequality, the diameter of the whole space is at most D + 2r. The converse does not hold: an example of a metric space that is bounded but not totally bounded is \(\R^2\) (or any other infinite set) with the discrete metric.

Compactness is a topological property which generalizes the properties of a closed and bounded subset of Euclidean space. There are several equivalent definitions of compactness in metric spaces:

1. A metric space M is compact if every open cover has a finite subcover (the usual topological definition).
2. A metric space M is compact if every sequence has a convergent subsequence. (For general topological spaces this is called sequential compactness and is not equivalent to compactness.)
3. A metric space M is compact if it is complete and totally bounded. (This definition is written in terms of metric properties and does not make sense for a general topological space, but it is nevertheless topologically invariant since it is equivalent to compactness.)

One example of a compact space is the closed interval [0, 1].

Compactness is important for similar reasons to completeness: it makes it easy to find limits. Another important tool is Lebesgue's number lemma, which shows that for any open cover of a compact space, every point is relatively deep inside one of the sets of the cover.

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.

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 if for every pair of points x and y in M₁, \[d_2(f(x),f(y))=d_1(x,y).\]

It follows from the metric space axioms that a distance-preserving function is injective. A bijective distance-preserving function is called an isometry. One perhaps non-obvious example of an isometry between spaces described in this article is the map \(f:(\R^2,d_1) \to (\R^2,d_\infty)\) defined by \[f(x,y)=(x+y,x-y).\]

If there is an isometry between the spaces M₁ and M₂, they are said to be isometric. Metric spaces that are isometric are essentially identical.

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 M₂, 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 M₁, the sequence \(f(x_1),f(x_2),\ldots\) converges to the point f(x) in M₂.

(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 M₁ and every ε > 0 there exists δ > 0 such that for all y in M₁ we have \[d_1(x,y) < \delta \implies d_2(f(x),f(y)) < \varepsilon.\]

A homeomorphism is a continuous bijection whose inverse is also continuous; if there is a homeomorphism between M₁ and M₂, 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, \(\R\) is unbounded and complete, while (0, 1) is bounded but not complete.

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 M₁ 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 M₁ to Cauchy sequences in M₂. 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 M₁ is compact, then every continuous map is uniformly continuous. In other words, uniform continuity cannot distinguish any non-topological features of compact metric spaces.