Re: Uniform spaces

Re: Uniform spaces

[Peter Freyd]

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

Re: Uniform spaces

[Oswald Wyler]

On Wed, 27 Aug 2003, Tom Leinster wrote:

> Date: Wed, 27 Aug 2003 16:51:55 +0200 (CEST)
> From: Tom Leinster <leinster@ihes.fr>
> To: categories@mta.ca
> Subject: categories: Uniform spaces
> 
> 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

Hi Tom,

The category UNIF of uniform spaces, without a separation axiom, is
topological over sets, and hence complete and cocomplete, with concrete
limits and colimits.  UNIF is not cartesian closed.

Cook and Fischer, Math. Ann. 173 (1967), 290-306, defined uniform convergence
structures of a set X as sets \scrF of filters on XxX satisfying five
axioms.  With the obvious definition of uniform continuity, sets with a
uniform convergence structure in this sense form a topological category
over sets, but Gazik, Kent and Richardson in Bull.Austral.Math.soc 11 (1974),
413-424, showed that this category is not cartesian closed.

In LNM 378, 591-637, I replaced the Cook-Fischer axiom that the principal
filter generated by the diagonal of XxX is in \scrF by the less demanding
axion that the principal filter generated by (x,x), for every x \in X,
is in \scrF.  This is now part of the accepted definition of uniform
convergence spaces.  In Bull.Austral.Math.Soc. 15 (1976), 461-465 my
student R.S. Lee showed that the category of uniform convergence spaces
with this definition is cartesian closed; this is not the cartesian closed
hull of UNIF.

For quasitoposes, we must go to semiuniform spaces which have partial
morphisms -- relations (m,g) with m an embedding -- represented by
one-point extensions.  Semiuniform convergence spaces and their uniformly
continuous maps form a quasitopos, but not the quasitopos hull of UNIF.
This has been determined by Adámek and Reiterman, The quasitopos hull of
the category of uniform spaes -- a correction, in the journal Topology
and its Applications.

For more information and literature, see my book Lecture Notes on Topoi
and Quasitopoi.

Oswald Wyler

Re: Uniform spaces

[Robert L. Knighten]

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

Re: Uniform spaces

[Michael Barr]

Any study of the category must begin with Isbell's wonderful book on the
subject.  Although John's exposition could be difficult, it was not so in
that book.  I don't recall about limits and colimits (but they ought to be
easy), but there is a lot of discussion of internal homs (which do not
always exist and are not symmetric when they do).  The category is not
cartesian closed.  I am pretty sure the forgetful functor to Top has a
left adjoint and therefore preserves limits.  It preserves sums for sure,
but not coequalizers since a quotient space of a hausdorff uniform space
can be hausdorff without being completely regular.  At least, that is what
I think I remember.

Michael

On Wed, 27 Aug 2003, Tom Leinster wrote:

> 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
>
>
>
>

Uniform spaces

[Tom Leinster]

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