Tom Leinster asks: Does anyone know of any account of the basic properties of the category of uniform spaces? I'm after things like (co)limits, cartesian closure, and (co)limit-preservation by the forgetful functor to Top. The place to start: Isbell, J. R. Uniform spaces. Mathematical Surveys, No. 12 American Mathematical Society, Providence, R.I. 1964 xi+175 pp. 54.30 The author gives an excellent introduction to recent results in uniform spaces, and, to a lesser extent, proximity spaces, especially the dimension theory of uniform spaces. The contents are roughly as follows. Chapter I: Metric and uniform spaces from the point of view of coverings, with uniform continuity and normal families of coverings. The entourage point of view is given in a problem at the end. Chapter II: Sums, products, subspaces and quotient spaces of uniform spaces, viewed from the vantage point of category theory. In addition, the completion and various compactifications of uniform spaces are discussed. Proximity theory is introduced briefly, as well as hyperspaces, i.e., the spaces of closed subsets of uniform spaces. Hyperspaces are treated by means of entourages. Chapter III: The functor $U(X,Y)$, the uniform space of all uniformly continuous functions from a uniform space $X$ to a uniform space $Y$ is defined, also by means of entourage, and an associated theory of injective spaces is developed. Next, equi-uniform continuity and semi-uniform products, and the chapter closes with the Ascoli theorem. Chapter IV: The metric topology is defined on (possibly infinite) simplicial complexes. Nerves of covers and canonical maps are defined, and results obtained on embedding uniform complexes in Euclidean spaces. Finally, inverse limits for uniform spaces are defined and developed, in the problems as well as in the text. Chapter V: Relations between the uniform dimension of a uniform space $X$ and the dimensions of subspaces and compactifications of $X$ are obtained. The concept of an ANRU, or uniform absolute neighborhood retract, is used to obtain some results on the extension of uniform maps and uniform homotopies, and a characterization of uniform dimension in terms of the extendibility of uniformly continuous maps of subspaces to $n$-spheres. The theory is then specialized to metric spaces. Chapter VI: Dimension-preserving compactifications of uniformizable topological spaces are considered relative to four distinct definitions of topological dimension. Useful examples are given of inequalities between the various dimensions. Some results on separable metric spaces and on Freudenthal compactifications of rim-compact spaces follow. Chapter VII: Except for a restriction on the cardinality of $X$, related to the problem of Ulam on "measurable cardinals", the author proves the Shirota theorem, essentially "that every topological space admitting a complete uniformity is a closed subspace of a product of real lines". Several more results on fine spaces are given, where a fine space is a uniform space whose uniformity is the finest compatible with its topology, among them a corollary of a theorem of Glicksberg's, that a product of fine spaces is fine if it is pseudo-compact. Chapter VIII: Several more results are given on the various dimensions for uniform spaces, mainly inequalities and sum theorems, together with a proof that the principal definitions coincide in the case of a separable metric space. An appendix follows which gives, among other things, a characterization of the real line in terms of uniformities. The author has an informal approach which brings out the main points well, and the discussion and problems are varied and interesting. Many open questions are mentioned, both large and small, and several research problems set, dealing with general questions of the structure of the theory and its extension. Three small points might be raised. First, Weil discussed coverings in his monograph, which antedates Tukey's, and chose the more algebraic approach of entourages. Somewhat more attention might have been given to his approach. Second, some more specific references to recent work relating dimension theory and algebraic topology would be useful. Third, notation indicating the chapter number on each page would have been useful, in view of the fact that the book will probably be a valuable reference for years to come. \{The author has forwarded the following corrections: Remarks about the Sierpi.'nski universal curve, page 122, are incorrect. The indications that Exercise II.4 and Theorem III.15 are not used are incorrect: these are page 32, page 41, and the places where they are used are III.6--7 and VII.1, respectively. The list of new results in Chapter VII (page iv) should not include VII.31. The main result of Reichbach [1], cited on page 12, is in Mostowski [Fund. Math. 29 (1937), 34--53]. The reference to Alfsen-Njestad [1], page 34, should be supplemented by reference to V. Poljakov [Dokl. Akad. Nauk SSSR 154 (1964), 51--54; MR 28 #582]. The reference (page 121) to Smirnov [7] for VI.16 should be Smirnov [ibid. 117 (1957), 939--942; MR 20 #276].\} Reviewed by M. A. Geraghty American Mathematical Society American Mathematical Society 201 Charles Street Providence, RI 02904-6248 USA (c) Copyright 2003, American Mathematical Society

Tom Leinster writes:

Hello,

Does anyone know of any account of the basic properties of the category of uniform spaces? I'm after things like (co)limits, cartesian closure, and (co)limit-preservation by the forgetful functor to Top. Bourbaki gets me some of the way, but his decision not to use categorical language and the resulting circumlocutions make it a struggle.

Thanks, Tom

It was written fairly early in the development of the category theory, but John R. Isbell Uniform Spaces Mathematical Surveys Number 12 American Mathematical Society xi+175pp, 1964 (Providence, Rhode Island) covers much of the territory and definitely with a categorical perspective. -- Bob -- Robert L. Knighten Robert@Knighten.org