Metric space

A set $\mathbb X$ together with a metric $\rho$. The set-theoretic approach to the study of figures (spaces) is based on the study of the relative position of their elementary constituents. A fundamental characteristic of the relative position of points of a space is the distance between them. This approach leads to the idea of a metric space, first suggested by M. Fréchet [2] in connection with the discussion of function spaces. It turns out that sets of objects of very different types carry natural metrics. As metric spaces one may consider sets of states, functions and mappings, subsets of Euclidean spaces, and Hilbert spaces. Metrics are important in the study of convergence (of series, functions) and for the solution of questions concerning approximation.

The development of the theory of metric spaces has proceeded in the following main directions.

General theory of metric spaces.

Here one studies properties of metric spaces which are invariant relative to isometries: one-to-one and onto mappings which preserve distance (cf. Isometric mapping). Such properties include completeness, boundedness, total boundedness, and widths. Properties of this type are called metric.

Topological theory of metric spaces.

Its subject is the properties of metric spaces which are preserved under homeomorphisms (cf. Homeomorphism). Among these are compactness, separability, connectedness, the Baire property, and zero dimensionality. Properties of this type are called topological.

Theory of spaces on which a metric has been given that is compatible with some additional algebraic structure (for example, a vector space or a group). Here one is concerned with Euclidean spaces, pre-Hilbert and Hilbert spaces (cf. Hilbert space) (of any weight), Banach spaces, Banach algebras, Banach lattices (cf. Banach space; Banach algebra; Banach lattice), and countably-normed spaces (cf. Countably-normed space). The facts available here are essentially connected with the discussion of important properties of metrics or norms, but the content, on the whole, belongs to the corresponding domains of algebra and functional analysis.

The discussion of particular metrics plays an important role in investigations in non-Euclidean geometries, differential geometry, mechanics, and physics. Here a central place is occupied by the notion of a Riemannian metric in a Riemannian space (see Riemannian geometry). A broader approach to the study of the surfaces and figures that arise in differential geometry is related to the concept of a -space, resulting from the addition of certain conditions to the metric axioms (see Geodesic geometry), which creates a basis for the discussion of geodesics in -spaces by ensuring their existence and nice properties. Typical here is the abandoning of the methods of differential calculus. In this connection it turns out that much of differential geometry is not connected with differentiability conditions, but is determined only by geometric axioms. Geodesic geometry is of interest not only as a generalization of Riemannian geometry, but also as an attempt to investigate geometric objects more geometrically, without using complicated computations.

In each set a metric can be defined by the following rule: if , and if . This is called the trivial metric. Each metric on a set gives rise, in a natural way, to a topology on . The concept of a topological space axiomatizes the relation of absolute nearness of a point to a set, whereas the concept of a metric space formalizes the notion of relative nearness of points. The distance of a point to a set in a metric space is defined as . A point is said to be absolutely near to a set if . The closure of in is the set of all points of absolutely near to . The topology in uniquely associated with this operation is called the topology generated in by the metric . The trivial metric leads to the trivial topology, in which all sets are closed.

In research on metric spaces (particularly on their topological properties) the idea of a convergent sequence plays an important role. This is explained by the fact that the topology of a metric space can be completely described in the language of sequences.

Let be a sequence of points in a metric space . It is said to converge to a point if for each there is an integer such that for all . The sequence is called fundamental if for each there is an integer such that for all .

An important metric property is completeness. A metric space is called complete if each fundamental sequence in it converges to a point of it. The space is always complete. Completeness of a metric space is not a topological property: A metric space homeomorphic to a complete metric space may be non-complete, for example, the real line with the usual metric is homeomorphic to the interval with the same metric; however, the first metric space is complete and the second is not.

Examples of complete metric spaces are Euclidean and Banach spaces. An important property of complete metric spaces, preserved under homeomorphisms, is the Baire property, on the strength of which each complete metric space without isolated points is uncountable. Therefore the usual topological space of the rational numbers is not generated by any complete metric. However, each metric space may be represented as a subset of some complete metric space by the standard construction of completion. Two fundamental sequences and in a metric space are called equivalent if

Let be the resulting collection of equivalence classes. A metric is introduced on by the rule: If and , , then

For , let , where for all . Then is a complete metric space and is an isometry of onto an everywhere-dense subset in ( is called the completion of ).

Related to the discussion of completion is the Lavrent'ev theorem on the extension of homeomorphisms. This theorem implies that the property of a metric space being a -set in its completion is a topological invariant (in contrast with the non-invariance of metric completeness itself relative to homeomorphisms).

Two metrics and on a set are called topologically equivalent if the topologies and generated by them coincide. On a finite set all metrics are equivalent; they generate the discrete topology. The Aleksandrov–Hausdorff theorem: A metric on a set is topologically equivalent to some complete metric if and only if is a -set in the completion of . In particular, the space of irrational numbers with the usual metric, relative to which it is not complete, is homeomorphic to the complete metric Baire space whose points are the infinite sequences of natural numbers, with distance given by , where is such that and for all .

The following example of a complete metric space is important: The space of all continuous functions on , with metric defined by the rule

for all . The space is separable — there is a countable everywhere-dense set in it (cf. Separable space). It turns out that each separable metric space is isometric to some subset of (the Banach–Mazur theorem). This result means that all metrics which generate separable topologies are obtained by restricting the natural metric on the set of continuous functions.

A subset of a complete metric space , equipped with the same metric (more precisely, its restriction to ), is a complete metric space if and only if is closed in .

There is a fundamental connection between the ideas of completeness and compactness for a metric space. Compactness of a metric space is equivalent to any of the following conditions: 1) any sequence in contains a convergent subsequence; 2) each countable open covering (cf. Covering (of a set)) of contains a finite subcovering; 3) in any open covering of there is a finite subcovering; 4) each decreasing sequence of non-empty closed sets in has a non-empty intersection; and 5) every closed discrete subset of is finite.

The simplest examples of compact metric spaces are: finite discrete spaces, any interval (together with its end points), a square, a circle, and a sphere. In general, a subset of the Euclidean space , with the usual metric, is compact if and only if it is closed and bounded.

The conditions listed are not all equivalent outside the class of metric spaces (see Compact space). H. Lebesgue (1911) established that for each open covering of a compact metric space there is a number such that each set of diameter is contained in some element of . This implies the fundamental property of compact metric spaces: Every continuous mapping of such a space into an arbitrary metric space is uniformly continuous (cf. Uniform continuity). Further, a metric space is compact if and only if each real-valued continuous function on it is bounded (and attains its least and greatest values).

Each compact metric space is complete, but the converse is false; the simplest example is an infinite discrete space with the trivial metric. However, the following characteristic property holds: A metric space is compact if and only if every metric space homeomorphic to it is complete.

It is intuitively clear that compactness implies, besides completeness, some sort of boundedness; this is confirmed by the consideration of compact subsets of . In general, a metric space is called bounded if there is a real number such that for all . Every compact metric space is bounded. The space is complete and bounded, but not compact if is infinite; thus, completeness and boundedness together are not sufficient for compactness in the class of metric spaces. In general, each metric on a set is topologically equivalent to some bounded metric, which is complete if the given metric is complete. In this connection, there is the important notion of total boundedness. A metric space is called totally bounded if for each there is a finite set such that for all . The set here is called an -net in . A metric space is compact if and only if it is complete and totally bounded, and is totally bounded if and only if it is isometric to a subset of some compact metric space. More precisely, total boundedness of a metric space is equivalent to compactness of its completion . Each subspace of a totally-bounded metric space is totally bounded. All totally-bounded metric spaces (in particular, all compact metric spaces) are separable and have a countable base. Compactness, in general, is not inherited by subsets; a set is relatively compact in a metric space if the closure of in is a compact metric space. If is complete, then relative compactness of a set in is equivalent to total boundedness of equipped with the metric .

An important role in functional analysis is played by a criterion for compactness of a set of continuous functions on in the metric space . This criterion is the following (the Arzelà–Ascoli theorem): A set is relatively compact in if and only if: 1) there is a number such that for all and all ; and 2) for each there is a such that for all and all for which .

A mapping of a metric space into itself is called a contraction if there is a real number such that

for all . An important theorem for complete metric spaces is the contracting- (contraction-) mapping principle (cf. also Contraction-mapping principle): For each such mapping of a (non-empty) complete metric space into itself there is precisely one fixed point.

The topological theory of metric spaces is significantly simpler than the general theory of topological spaces. Below the most important topological properties of metric spaces are given. Here one has in mind properties of the topology that is generated by the metric .

Each metric space is normal and even collectionwise normal (cf. Normal space). This permits the extension of continuous real-valued functions from closed subsets of a metric space to the whole space. A stronger result is: For each closed subset of a metric space there is a linear mapping of the space of all continuous real-valued functions on to the space of all continuous real-valued functions on such that (for any ) is an extension of and

(Dugundji's theorem). This theorem is related to Hausdorff's theorem on the extension of metrics: If a closed subspace of a metrizable space is already metrizable with a metric (generating the topology on as a subspace of ), then it is possible to extend to a metric on the whole of , generating the original topology on . Similar results are valid for totally-bounded metrics and complete metrics.

Research into the topological properties of metric spaces is, to a large extent, based on the following theorem of A.H. Stone: A metric space is paracompact, that is, any open covering has an open locally finite refinement (locally finite means that each point has a neighbourhood intersecting only a finite number of elements of , cf. also Paracompact space). The Nagata–Smirnov metrization criterion (see Metrizable space) is based on the paracompactness of metric spaces.

For a metric space there are important theorems on the equivalence of topological properties which are distinct in general topological spaces. Thus, the following cardinal-valued invariants coincide: the density, the character, the weight, the Suslin number, and the Lindelöf number (cf. also Cardinal characteristic). For metric spaces countable compactness, pseudo-compactness and compactness are equivalent. For metric spaces the dimensions (the covering dimension) and (the large inductive dimension) coincide, and for separable metric spaces the small inductive dimension coincides with and (see Dimension theory).

Each metric space is star normal; any open covering of has an open star refinement , that is, for each point there is a containing every for which . Related to this theorem is the following metrizability criterion (Stone–Arkhangel'skii): A regular space is metrizable by a totally-bounded metric if the space has a countable base. But even a countable regular space need not be metrizable. The simplest example is obtained by adjoining to the discrete natural numbers one exterior point from its Stone–Čech compactification. The criterion for metrizability of a metric space by a complete metric is unexpected: It is necessary and sufficient that be a -set in some (and then in any) Hausdorff compactification of . However, Hausdorff compactifications of metric spaces carry complete information on the topology of the latter; this is clear from Čech's theorem: Metric spaces are homeomorphic if and only if their Stone–Čech compactifications are homeomorphic.

A metric space need not have a countable base, but it always satisfies the first axiom of countability: it has a countable base at each point. In addition, each compact set in a metric space has a countable base. Moreover, in each metric space there is a base such that each point of the space belongs to only countably many of its elements — a point-countable base, but this property is weaker than metrizability, even for paracompact Hausdorff spaces. Regular separable spaces satisfying the first axiom of countability need not be metrizable.

The condition for metrizability of a separated topological group is easily found: It is necessary and sufficient that the space of the group satisfies the first axiom of countability; there are then both left-invariant and right-invariant metrics generating the topology.

Connected with each metric space , in a standard way, there is another metric space, namely the space of its non-empty bounded closed subsets with the Hausdorff metric, which is defined as follows:

The space is isometric to a closed subspace of . If is complete, then is complete. But topological equivalence of two metrics and given on does not imply, in general, that the corresponding Hausdorff metrics and are topologically equivalent.

A continuous image of a metric space need not be homeomorphic to any metric space, even when the Hausdorff separation axiom is satisfied. This also applies to quotient spaces of metric spaces. For example, if in the plane one shrinks a fixed line to a point, taking as individual elements of the decomposition all the points of the plane not on the line, then one obtains a non-metrizable separable space, at whose special point the first axiom of countability is not satisfied. There is a general criterion for the metrizability of a quotient space of a metric space (see [6]). In particular, the quotient space associated with a continuous decomposition of a metric space into compact sets is always metrizable. Every Hausdorff space which is a continuous image of a compact metric space is metrizable and compact. This is a particular case of a general proposition on the non-increase of the weight of a topological space under a continuous mapping into a compact set. But even when the image of a metric space is metrizable, the metric realizing the metrization of need not be obtained from by means of any formula. Instead of a metric on , with respect to it is natural to define a function by means of the rule: For , is equal to the distance in the sense of between the inverse images of the points under the mapping under discussion. Often (for example, if is the decomposition space of a metric space into compact sets) is closely related to the topology of and is symmetric. This means that for all , if and only if . The symmetric relation thus defined almost never satisfies the triangle axiom (cf. Metric), but if the decomposition into compact sets is continuous, then has topological properties which successfully replace the triangle axiom and guarantee the metrizability of the image by a "true" metric. A topological space which is the image of a metric space under a continuous open and closed mapping is itself homeomorphic to a metric space. However, under continuous open mappings, metrizability is not always preserved: All spaces satisfying the first axiom of countability, and only they, are the images of metric spaces under continuous open mappings.

Among the generalizations of metric spaces the most important are pseudo-metric spaces, spaces with a symmetric relation and spaces with -metrics [7]. These are defined axiomatically by a natural weakening of the axioms of a metric space. However, the distance here, as usual, is expressed by a non-negative real number. One may consider generalized metrics with values in ordered semi-groups, semi-fields, etc. (see [8]). In this way a generalized metrization of an arbitrary completely-regular space can be obtained.

A fundamental generalization of the concept of a metric space is the notion of a uniform space. Further, there are purely topological extensions of the class of metric spaces, among which are the important classes of spaces with a uniform base, Moore spaces, feathered and paracompact feathered spaces, and lattice spaces (cf. Moore space; Feathered space). The class of paracompact spaces is too broad a generalization of the class of metric spaces, because paracompactness is not even preserved under finite products. On the contrary, the class of paracompact feathered spaces is a successful simultaneous generalization of the class of spaces homeomorphic to metric spaces and the class of compact spaces. In another direction the idea of a metric generalizes to a -metric and a -metric [4]. The concept of a statistical metric space, introduced by K. Menger, turns out to be topologically equivalent to the idea of a space with a symmetric relation.

References

[1]	P.S. Aleksandrov, "Einführung in die Mengenlehre und die allgemeine Topologie" , Deutsch. Verlag Wissenschaft. (1984) (Translated from Russian)
[2]	M. Fréchet, "Sur quelques points du calcul fonctionnel" Rend. Circ. Mat. Palermo , 22 (1906) pp. 1–74 Zbl 37.0348.02
[3]	A.V. Arkhangel'skii, V.I. Ponomarev, "Fundamentals of general topology: problems and exercises" , Reidel (1984) (Translated from Russian) MR785749 Zbl 0568.54001
[4]	E.V. Shchepin, "Topology of limit spaces of uncountable inverse spectra" Russian Math. Surveys , 31 : 5 (1976) pp. 155–191 Uspekhi Mat. Nauk , 31 : 5 (1976) pp. 191–226 Zbl 0356.54026
[5]	R. Engelking, "General topology" , Heldermann (1989) MR1039321 Zbl 0684.54001
[6]	A.V. Arkhangel'skii, "Factor-mappings of metric spaces" Soviet Math. Dokl. , 5 : 2 (1964) pp. 368–371 Dokl. Akad. Nauk SSSR , 155 : 2 (1964) pp. 247–250 Zbl 0129.38104
[7]	S.I. Nedev, "-metrizable spaces" Trans. Moscow Math. Soc. , 24 (1971) pp. 213–247 Trudy Moskov. Mat. Obshch. , 24 (1971) pp. 201–236 MR0367935 Zbl 0417.54008 Zbl 0295.54039 Zbl 0257.54025 Zbl 0255.54026 Zbl 0255.54025 Zbl 0248.54034 Zbl 0246.54035 Zbl 0242.54030 Zbl 0242.54029
[8]	M.Ya. Antonovskii, V.G. Boltyanskii, T.A. Sarymsakov, "An outline of the theory of topological semi-fields" Russian Math. Surveys , 21 : 4 (1966) pp. 163–192 Uspekhi Mat. Nauk , 21 : 4 (1966) pp. 185–218
[9]	S.I. Nedev, M.M. Choban, "-metrics and proximity spaces. Metrization of proximity spaces" Serdica , 1 (1975) pp. 12–28 (In Russian) MR0394579

Comments

The trivial metric is also called the discrete metric. A fundamental sequence is also called a Cauchy sequence. Star-normal spaces are also called fully normal.

There are fairly obvious numerical invariants of metric spaces such as width (diameter) and (various kinds of) dimension. A rather more hidden numerical invariant is the Gross dispersion number or rendezvous number, whose existence and uniqueness is guaranteed by the following theorem, [a12]. Let be a compact connected metric space; then there is a unique number such that for all and all sets of points there is a point such that

(a1)

Some examples are as follows. If is a ball of radius in Euclidean -space, then ; if is the -dimensional sphere of unit diameter in Euclidean -space, then, [a15],

where is the gamma-function; if is an equilateral triangle in , then . The theorem guaranteeing the existence of generalizes in two directions. First, can be replaced by any symmetric function (where symmetric means ), [a13], and further the average on the left of (a1) can be replaced by an integral, [a14]. Thus, for a compact connected Hausdorff space and a symmetric function there exists a unique real number such that for any regular Borel probability measure on there is a point such that

The metric spaces on which every continuous function (to any metric space, or just to the real line) is uniformly continuous are studied in [a2]. The simplest description is this: There is a compact subset such that the complement of any neighbourhood of is discrete.

Beyond the completion of a metric space is the injective envelope . In general, is not dense in (e.g., if is a circle, is infinite dimensional), but is an essential extension of ; this means that a non-expansive mapping from to any metric space whose restriction to preserves all distances must preserve all distances in (non-expansive means that for all ). is characterized, up to a unique isometry, as an essential extension of which has no further essential extension. This is equivalent to injectivity in the sense of extendability of mappings, and also to the following Helly-type property: Any family of spherical neighbourhoods satisfying the consistency conditions , for all , has a common point. Because of this equivalence, the injective metric spaces are also called hyperconvex. See [a1], [a5].

The injective envelope of a real Banach space is itself a real Banach space in a unique compatible way. This result is known only from a highly non-constructive proof combining H. Cohen's construction of relative injective envelopes in the category of real Banach spaces [a3], the Aronszain–Panitchpakdi theorem that an injective real Banach algebra is an injective metric space [a1], and the Mazur–Ulam theorem that every isometry of real Banach spaces is affine [a9]. Compare [a6].

Injective spaces support a much stronger fixed-point theorem than the contraction-mapping theorem: Every non-expansive mapping of a bounded injective metric space into itself has a fixed point (the Sine–Soardi theorem). However, the extensive development of fixed-point theory of non-expansive mappings has been done mostly in the important special case of convex subsets of Banach spaces. A survey of it, up to 1980, is in [a8]. Compare [a7].

It should be noted that the Stone–Arkhangel'skii metrization criterion involves A.H. Stone, who also proved the paracompactness of metric spaces. In the Stone–Čech compactification, it is M.H. Stone.

It was noted above that two topologically-equivalent metrics on a space do not, in general, give topologically-equivalent Hausdorff metrics on the hyperspace . In fact, they do so if and only if they are uniformly equivalent [a4].

References

[a1]	N. [N. Aronszain] Aronszajn, P. Panitchpakdi, "Extensions of uniformly continuous transformations and hyperconvex metric spaces" Pacific J. Math. , 6 (1956) pp. 405–439 MR0084762
[a2]	M. Atsuji, "Uniform continuity of continuous functions of metric spaces" Pacific J. Math. , 8 (1958) pp. 11–16 MR0099023 Zbl 0082.16207
[a3]	H. Cohen, "Injective envelopes of Banach spaces" Bull. Amer. Math. Soc. , 70 (1964) pp. 723–726 MR0184060 Zbl 0124.06505
[a4]	D. Hammond Smith, "Hyperspaces of a uniformizable space" Proc. Cambridge Philos. Soc. , 62 (1966) pp. 25–28
[a5]	J. Isbell, "Six theorems about injective metric spaces" Comment. Math. Helv. , 39 (1964) pp. 65–76 MR0182949 Zbl 0151.30205
[a6]	J. Isbell, "Three remarks on injective envelopes of Banach spaces" J. Math. Anal. Appl. , 66 (1969) pp. 301–306 MR0251512 Zbl 0206.42201
[a7]	V.I. Istraţescu, "Fixed point theory" , Reidel (1981) MR0620639 Zbl 0465.47035
[a8]	W.A. Kirk, "Fixed point theory for nonexpansive mappings" , Fixed Point Theory , Lect. notes in math. , 886 , Springer (1981) pp. 484–505 MR0643024 Zbl 0479.47049
[a9]	S. Mazur, S. Ulam, "Sur les transformations isométriques d'espaces vectoriels" C.R. Acad. Sci. Paris , 194 (1932) pp. 946–948
[a10]	J. Dugundji, "Topology" , Allyn & Bacon (1966) MR0193606 Zbl 0144.21501
[a11]	J.L. Kelley, "General topology" , Springer (1975) MR0370454 Zbl 0306.54002
[a12]	O. Gross, "The rendezvous value of a metric space" M. Dresher (ed.) L.S. Shapley (ed.) A.W. Tucker (ed.) , Advances in game theory , Princeton Univ. Press (1964) pp. 49–53 Zbl 0126.16401
[a13]	W. Stadje, "A property of compact connected spaces" Arch. Math. , 36 (1981) pp. 275–280 MR0620518 Zbl 0457.54017
[a14]	J. Cleary, S.A. Morris, D. Yost, "Numerical geometry—numbers for shapes" Amer. Math. Monthly , 93 (1986) pp. 260–275 MR0835294 Zbl 0598.51014
[a15]	S.A. Morris, P. Nickolas, "On the average distance property of compact spaces" Arch. Math. , 40 (1983) pp. 459–463 MR707736 Zbl 0528.54028