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[Texts]## 54E: Spaces with richer structures especially metric spaces |

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

- positivity (d(x,x)=0 but d(x,y) > 0 if x is distinct from y)
- symmetry (d(x,y)=d(y,x)), and
- the triangle inequality (d(x,y) is at most d(x,z)+d(z,y) for every z).

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.

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.

- 54E05: Proximity structures and generalizations
- 54E15: Uniform structures and generalizations
- 54E17: Nearness spaces
- 54E18: p-spaces, M-spaces, sigma-spaces, etc.
- 54E20: Stratifiable spaces, cosmic spaces, etc.
- 54E25: Semimetric spaces
- 54E30: Moore spaces
- 54E35: Metric spaces, metrizability
- 54E40: Special maps on metric spaces
- 54E45: Compact (locally compact) metric spaces
- 54E50: Complete metric spaces
- 54E52: Baire category, Baire spaces
- 54E55: Bitopologies
- 54E70: Probabilistic metric spaces
- 54E99: None of the above but in this section

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.

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

- Statement of a couple of metrization theorems
- Distinguishing notions of equivalence among metrics.
- A shot at metrizing topological fields (and embedding them into R).
- Are all self-maps really contractions with respect to some metric?
- Making the space of continuous functions from R to R into a metric space
- A description of Blumberg's Theorem -- functions are always continuous on a dense set
- Is completeness a homeomorphic invariant? (no)
- Making the set of irrational numbers into a complete metric space.
- Non-sequence-based completion of a metric space [Ron Bruck]
- Characterization of compactness in metric spaces
- Discussion of dimensions of metric spaces and their products.
- Citations: into what Euclidean spaces can a metric space be embedded?
- An interesting example of computing Hausdorff dimension (and continued fractions!)
- What is the Hausdorff metric on the set of (closed) subsets of a space?
- Functions continuous precisely at the rationals? No (by the Baire Category Theorem).

Last modified 1999/05/12 by Dave Rusin. Mail: feedback@math-atlas.org