[The following does not rise even to the level of an essay-in-progress; more like a thought-in-progress, or even (saving your presence) a difficult bowel-movement. But the hordes of typist-elves in the cavernous warehouses of WDJ have yet to present anything brought to perfection this morning, and I wished not to disappoint the milling crowds that swarm this site each weekend, bringing the whole family, Sister Sue and Fido too, gawking at the glittering thoughtfronts -- the polemics, the poems, the darling little monostichs (these we can all afford) -- while shaking their heads sadly at the Trinitarian Minimalism and Cantorian Realism (out of our price-range) -- all save one diminutive child towards the back of the bunch, eyes riveted on the prize, instinct with penetrating understanding…]
We saw
here the dialectic of mathematical invention (not trying to be too Hegelian here -- think of it as an ensouled pendulum) whereby, beginning with the everyday world we live in -- I almost wrote ‘space’, but that would be to get ahead of our tale -- we
abstract from the clutter of minute-to-minute experience, and conceive of it all happening within a
space. We then
formalize that space with the Euclidean axioms. We then familiarize ourselves with this new mind-environment, solving tricky problems and whatnot for a couple of thousand years, then -- since we have long effectively been working in the World of the Unseen -- very lightly
generalize to Euclidean spaces of any finite dimension -- a bit of a stretch biologically, but where, mathematically, everything works pretty much as before.
Meanwhile independently, mathematical analysis had proceeded apace, not necessarily concerned with the geometrical substrate as such, but piling up its own increasingly intricate problematics. Then by an ideational leap which is of the essence of mathematics, and into which simply listening to lectures and slogging through the problem-sets at the end of the chapters, gives you no insight at all (executive summary: Mathematicians are like gods), a clutch of bold spirits, bearing in mind certain delicate problems such as infinite sequences of functions and their convergence, generalized the stage on which such pageants play out, from the Euclidean to the general topological. (The history has here been brutally telescoped.) Something of the sort was in any case needed to save the Euclidean picture itself, since infinite-dimensional spaces were now required (even by physics), and the finite-dimensional structures would not generalize in any straightforward way.
General topological spaces being a wildly assorted bag, various restrictions are put on them, for one purpose or another, to allow deduction and calculation. One of these is
metrizability, which we examined in the essay on
Urysohn. That has the advantage of preserving much of our hard-won familiarity with the Euclidean metric, while allowing a vast array of new metrics for particular purposes. (For example: the by-now-familiar Lorentz metric of Einsteinian spacetime. Once mind-boggling, yet now -- in this vaster context -- almost cuddly.) These in turn can be slightly re-generalized, by considering pseudometrics; or further regimented, with the concept of a
norm, which in turn may be relaxed into a seminorm; and so it goes.
~
A quite different and likewise fruitful generalization of metric spaces is the notion of a Uniform Space, introduced by algebraic geometer
André Weil, in “Sur les espaces à structure uniforme et sur la topologie générale” (reprinted in volume I of his
Collected Papers as [1937]). He broaches it with a bang:
La notion de distance est utilisée dans de nombreux travaux de topologie, [mais] l’on s’explique mal qu’elle soit venue à jouer un pareil rôle dans une branche des mathématiques où elle n’est, à proprement parler, qu’une intruse…
On voit apparaître ici cette hypothèse du dénombrable (dite aussi, on ne sait pourquoi, de séparabilité), malfaisant parasite qui infeste tant de livres … dont il affaiblit la portée tout en nuisant à une claire compréhension des phénomènes. … La conscience d’un mathématicien, s’il en possède [!], doit répugner à faire intervenir une hypothèse superflue …
Strong words ! The notion of metric, he claims, is not simply too restrictive, but is the wrong sort of notion for topology -- a cuckoo’s-egg in the nest. And indeed, minus the polemics, James Dugundji makes the same point (Topology, p. 200):
A metric … can be regarded a providing a measure of nearness that is applicable throughout the space … This notion of uniform smallness is not a topological concept : equivalent metrics specify different sets as being equally small.
… Notice that, even in metric spaces, a continuous map may be uniformly continuous if one pair of metrics is used, but not uniformly continuous when another pair of equivalent metrics is used; uniform continuity is therefore not a topological concept.
(“Equivalent” metrics in the sense that they generate the same roster of open sets, which define the topology.)
Contrast a different -- and very fruitful -- restriction on general topological spaces, that of being
compact Hausdorff. This notion
is strictly topological in spirit.
Footnote: For another instance of Gallic arithmophobia, cf. the remarks of Weil’s countryman Jean Dieudonné, in Foundations of Modern Analysis (1960), p. 141:
The fundamental idea of Calculus [is] the “local” approximation of functions by linear functions. In the classical teaching of Calculus, this idea is immediately obscured by the accidental fact that, on a one-dimensional vector space, there is a one-to-one correspondence between linear forms and numbers, and therefore the derivative at a point is defined [horresco referens !] as number instead of a linear form.
In defense of Sir Isaac Newton, it must be observed, that our worthy ancestor was quite understandably interested in how fast something was going, at each moment: to answer which question, he needed to invent the differential calculus. Dieudonné, from the vantage point of centuries of progress, is looking ahead to function-spaces and dense subsets of special functions and like that.
~
The passages immediately above evoke, unbidden, an untoward echo characteristic of their times (the Thirties; the Sixties): “unAmerican” and (failure to adhere to) “Chairman Mao’s Correct Line”. But “topological” is not an all-or-nothing concept; and we return to sanity with jolly John Kelly (General Topology), in the chapter titled “Uniform Spaces”:
We deduce from a topological premise (that the space is compact) a non-topological conclusion (that a function is uniformly continuous). This chapter is devoted to a study of quasi-topological results of this sort.
Even more telling is the remark by George Simmons, author of the superbly pedagogical Introduction to Topology and Modern Analysis (1963):
Some writers deal with the theory of metric spaces as if it were merely a fragment of the general theory of topological spaces. This practice is no doubt logically correct, but it seems to me to violate the natural relations between these topics, in which metric spaces motivate the more general theory.
Thus, it is scarcely fair, or psychologically realistic, to denounce the notion of metric as an “intruder” in topology, as Weil does. Similarly: you shouldn’t start off with categories and functors before learning about ordinary numbers and sets, even if categories prove ultimately more foundational.
That said, there does come a point where actual everyday examples
impel one to consider such things as convergence and compactness in a setting more general than a metric
space. As: pointwise convergence, which is a perfectly familiar non-exotic
sort of convergence, but which cannot be seen as convergence with respect to a metric.
~ ~ ~
We have thus seen uniform space as a gentle generalization of metric spaces. Since the point of the latter is often concerned largely with matters of limits and convergence, all we really need to know is what it means to get “closer and closer”; we don’t need to put a number on how close, each step of the way. This aspect was highlighted by André Weil, when he debuted the idea of uniform spaces, as a kind of intellectual hygiene. But in practice, quite as important to the introducer of uniform spaces is their natural application to topological groups, which come ready-made with a structure amenable to notions of nearness.
But there is more. John Kelley, in his General Topology (1955), who devotes an entire chapter to uniform spaces, writes:
It should be emphasized that this is by no means the only framework in which uniformity can be studied. It is possible to study a set X together with a distinguished family of pseudo-metrics for X, or to distinguish a collection of covers of X where are to be uniform covers (roughly in the sense of the Lebesgue covering lemma). One may also consider “metrics” with values in a structure less restricted than that of the real numbers. All of these notions are essentially equivalent.
Such a situation illustrates a recurring intellectual theme of this series of essays (with both philosophical and mathematical applications), tagged as “
Rome by different roads”. There is a section on this notion in our essay
Consilience in mathematics (indeed, in one sense, the entire notion of consilience in general is related to this idea).
