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54E: Spaces with richer structures especially metric spaces


Introduction

In this section of Topology, one considers sets X with an additional structure which in turn implies a topology on X. Although of course the tools of General Topology apply to these spaces, one expects more specific answers to topological questions about these spaces, yet one may also ask new questions which make explicit reference to the additional structure.

The most important kinds of structure considered here are those of a metric or related constructs such as a uniform structure. (Cellular or simplicial complexes, as well as manifold-like structures are also critically important but are parts of other subdisciplines of topology; see section 57.)

There is of course no end to the sets of additional structures which may be placed on a topological space. Some of them are introduced because they arise naturally in applications; in general, these structures warrant continued attention over time. Others are introduced for technical reasons: the extra structures may make some proofs easier, or they clarify the most general circumstances under which a proof may hold. Some of these are subsequently neglected when it becomes clear that the optimal set of axioms has not been chosen.

A metric on a set X is a real-valued function d on X x X which satisfies

that is, d serves as a distance. A set X with a metric on it is called a metric space since the collection of unions of balls is a topology on X.

A major theme in research is to investigate the influence a metric has on the underlying topology. For example, one has characterizations of compactness in metric spaces (e.g. the Heine-Borel and Arzelà-Ascoli theorems). In some cases the influence is completely determining, that is, metric spaces are always Hausdorff (even normal), first countable, and paracompact (Stone's theorem). Indeed, one may ask for sets of topological conditions on a topological space which imply the topology is consistent with a metric, that is, one may ask about metrizability. Theorems of this type (e.g. those of Urysohn, Nagata, Smirnov, Morita, Stone,..) are remarkable in that they create a function (the metric) comparing spaces to a specific space (the real line) just from internal properties of the space. (Mary Ellen Rudin has been quoted as having said that "metrization is the heart and soul of general topology"!)

Other avenues of investigation in metric spaces ask questions which are not detected by the topology. Principal among these is the concept of completeness (that is, whether all Cauchy sequences converge in the space). There are well-known characterizations of complete spaces and constructions of completions of general metric spaces. Applications of completeness (e.g. the Baire Category Theorem) lead to useful existence results in analysis. (Indeed, one may generalize this, and consider Baire spaces -- spaces in which the conclusion of that theorem holds.)

Within a metric space it is reasonable to ask about convergence of a sequence, and in particular one may ask questions about dynamics (e.g. convergence of images of iterates of a map). This is the also setting for a number of variations of the Banach contraction mapping principle. These, like many questions about metric spaces, take on particular significance when applied to metric spaces of particular importance, such a Euclidean spaces, spaces of functions (especially Banach spaces and other topological vector spaces), spaces of compact sets (the Hausdorff metric of fractals), p-adic metric spaces, and specific metric spaces used in constructions and examples (e.g the Hilbert cube, and some exotic subspaces of R^n such as the sin(1/x) curve).

With a metric in place it is reasonable to ask questions of a geometric nature, although a metric per se allows only an analogue to lengths, not to angles. Still, one may pursue questions about isometries, embeddings, and isometry groups. It is also reasonable to pursue measure-theoretic questions, to define dimensions and volumes in arbitrary metric spaces. Again, the most dramatic results are usually associated with familiar metric spaces such as R^n, but the techniques are usually valid in the broad setting.

The utility of metric spaces fosters the consideration of related spaces. Among those mentioned in the MSC we observe semimetric spaces (topological spaces whose topology is given by the balls with respect to a semimetric -- a distance function not meeting the triangle inequality); cosmic spaces (continuous image of a separable metric space), and probabilistic metric spaces. Moore spaces and stratifiable spaces are roughly, those whose open sets are unions of special families of open sets; the question of whether or not these are metrizable involves subtle themes in axiomatic set theory! Gauge structures replace the strict positivity of a metric with the collective separating power of a set of distance functions.

If topological spaces can be described as those sets on which one may discuss continuity, then uniform structures are those which all discussion of uniform continuity. This includes not only metric spaces but also topological groups and topological vector spaces. (These subjects are given their own headings in the MSC.) The axiomatization is due to André Weil. Uniform structures are more general than metrics; indeed, one may prove a space is uniformizable iff it is completely regular. Uniformity, too, admits generalizations (e.g. nearness spaces weaken the start-refinement condition of uniform spaces).

Other structures include proximity structures (relations among subsets of the space comparable to being overlapping); bitopologies (spaces with two topologies; equivalently, a topology on XxX -- Kelly 1963); ... Strictly speaking each of these forms a category with an underlying ("forgetful") functor to the category of topological spaces. One can investigate the full range of categorical constructions (subspaces, products, maps between spaces, functors to other categories especially forgetful functors, ad infinitum). In general these less well-known structures are used primarily for their simplicity in formulating or proving results of importance in the more traditional categories of metric spaces (say) or general topological spaces.

History

Applications and related fields

The other most common spaces with extra structure are the ones treated in 57: Manifolds and Cell Complexes. Many are metric spaces in addition to being manifolds (say).

Topological vector spaces are studied in 46: Functional Analysis. Topological groups are in section 22.

Topics involving not only a metric (length) but angles require Geometry, although in the broadest sense these topics involve inner product spaces (especially Hilbert spaces), which are subjects in 15: Linear Algebra. These comments apply in particular to questions regarding the metric spaces R^n.

General metric-space topics (such as convergence or completion) when applied to R^n or functions between subspaces of R^n are typically considered part of the appropriate portion of analysis (e.g. 26: Real Analysis).

Topics in dynamical systems on metric spaces are typically in 58: Global Analysis.

Non-Archimedean metrics (such as the p-adic metrics) are useful tools in 11: Number Theory.

A metric in the sense of differential geometry is only loosely related to the concept of a metric in a metric space.

Subfields

CW complexes, once section 54E60, have been moved to section 57.

Parent field: 54: General Topology

Browse all (old) classifications for this area at the AMS.


Textbooks, reference works, and tutorials

The theory of metric spaces is almost always presented with an eye towards its connections either with general topology or with analysis; this is true both at the beginning undergraduate level and at advanced levels.

A fairly nice balance is achieved by Köhnen, Walter: "Metrische Räume, Ein Lehr- und Übungsbuch", Academia Verlag Richarz GmbH, Sankt Augustin, 1988. 534 pp. ISBN 3-88345-662-4, MR 90f:54001a,b (a solutions manual is available)

For an elementary undergraduate text, one cannot top the writing of Kaplansky, Irving, "Set theory and metric spaces", Chelsea Publishing Co., New York, 1977. 140 pp. ISBN 0-8284-0298-1, MR56 #5297

For consideration of other structures, see James, I. M., "Introduction to uniform spaces", Cambridge University Press, Cambridge, 1990. 148 pp. ISBN 0-521-38620-9, MR 91m:54033; and

Preuss, Gerhard, "Theory of topological structures", D. Reidel Publishing Co., Dordrecht-Boston, MA, 1988. 304 pp. ISBN 90-277-2627-2, MR89m:54014

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