Uniform Spaces

[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.

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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.

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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.

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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).

The Urysohn Metrization Theorem (concluded)

(The continuation to this.)


Is there any distinction between a metrizable space and a metric space?   Seen naively, it’s the difference between a barn that hasn’t been painted yet, and one that has.

            Mathematically, the difference is insignificant.  Notice how one of the statements of the theorem  quoted above  slurs over the distinction:

     A compact Hausdorff space that is second countable is a metric space.

There is no mathematically interesting category of metrizable spaces prior to actual imposition of some specific metric -- analogous, say, to entangled quantum particles prior to collapse of the wave-packet, which are very interesting indeed, both philosophically (EPR Theorem, Bell’s experiments) and practically (quantum computing, quantum cryptography).  (For a quick course in the Uncertainty Principle, click here.)  If there actually were an analogy, how neat it would be, since in both cases  the final step involves (in some sense) “measurement”.

            There is, though, a lesson here for our larger project of Cantorian Realism, and the ontology and epistemology of mathematical objects.   Thus, consider a space (given initially as a base set and a defined neighborhood-system) which, after fiddling awhile, we find to be regular and second-countable.  Aha, so it’s metrizable, though knowing this does not by itself hand us a workable metric;  we’ll have to see what works.   Here, clearly, the metaphor of the unpainted barn breaks down.   For if barns -- which we build -- were like mathematical objects -- which (it is our contention) we discover -- some of them would prove recalcitrant to painting -- purely and simply unpaintable;  much as the Long Line can never be metric, howsoever it twist and turn.   Further, some paintable barns would admit more than one hue of paint, though not indefinitely many.

For let us emphasize:  Being metrizable is not a property of a bare set, but of a topological space -- that is, a base set together with a roster of which subsets count as open -- this roster itself is referred to as the “topology”.  The question then is whether a metric can be defined on the base set that will induce that roster of open-sets.  We have already been given the open sets we’re ‘aiming for’;  if the metric fails to yield these, then it is not a metric for that topology.  If no metric yields the right open sets, then that space (with that topology) is not metrizable.

Example 1:  Take the real plane, R x R, and let the interior of circles (i.e., open discs) be a basis for the topology.  Now define a metric on this set such that d(x,y) = 1 for all pairs of distinct points in the set.  This metric induces a topology all right -- the discrete topology, in which every pointset is itself open -- but it is not the Euclidean topology;  no cigar.  (Note:  The discrete topology is that of Leibnizian monadology, where every man is an island unto himself.)  The space itself is metrizable, however;   just use the usual Euclidean metric.

Example 2:  Now take a countably-infinite product of the set of reals with itself, R x R x R …  (You can pronounce this “R to the omega”.)   Assign the usual product topology to this (in which all but finitely many of the projections of an open set onto the individual R’s  must be all of that R).  You can induce this topology via a modification of the uniform norm.   But now instead assign the box topology (in which there is no restriction on how many of the slices may be less than all of R).  No metric induces that topology.

            As Dauben reports, Cantor himself eventually discovered the strange gap between our meeting a mathematical object for the first time -- presumably full-blown, yet still partially inscrutable -- and any eventual fullness of understanding. “Cantor no longer assumed that every set is born well-ordered.”

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            Though the superficial similarity of the quantum case and the U.M.T.  doesn’t hold up, there does appear to be a rather arresting analogy with post-Chomskyan linguistics.

            The traditional view of language learning was that it involved general learning-strategies:  learning to make relative clauses was not radically different from learning your colors or the names of the kings of England (I caricature somewhat):  and just as different peoples conceive the color-palette in apparently incompatible ways, and the order of the kings might have been different (or no kings at all), so languages could differ indefinitely.

            Chomsky then challenged all this in ways much deeper and more philosophical than appeared to most people at first.   Many were surprised when, after laboring for a while at the forefront of fashionable linguistics, he out of the blue published a study of the time of Descartes, far outside the intellectual horizons of most of his followers.   But indeed, his project coheres, and always has.  Following his thought over the years, and finally getting the point, is a bracing intellectual experience.

            What initially attracted people was the positive expressive power in the slogan “Generative Grammar”;  yet very soon, those at the heart of the enterprise began to emphasize the theme of constraint. 

            In Chomsky’s view, as language-learners we must contend with certain hard (as in: hard-wired), quasi-algebraic parameters, each with a small finite range of possible values (often just two).  By our exposure to the particular ambient language in which we find ourselves, we (unconsciously) flip the various switches to their contingent, discovered position.   Certain combinations of settings will have further structural consequences.

            If we were as happily wired for topology as we are for language, we would meet a space, play with it in our cribs, learn in time what is the setting for its Separation parameter (T1, Hausdorf, regular, normal…), its Countability parameter (first-countable, second-countable, or neither) -- and having found that it is regular and second-countable, we would know it to be metrizable.

            Chomsky’s approach has been said, including by his fans, to involve an “innateness hypothesis”, a term at which he sometimes bridled.   And indeed, I called this roster of pre-existent parameters simply “hard”, where the imagery could be that of crystals (a full complement of Platonic solids, say) rather than that of a wiring diagram.  The default assumption in our scientific culture is, of course, that they reside on some gene or other;   but their actual nature renders problematic (not impossible) their visibility to the usual processes of Natural Selection.   (And again, to the puzzlement of his friends, Professor Chomsky never leapt with one bound onto the Darwin bandwagon.)

Also, if these parameters were coded for separately, one might expect a richer panoply of language-related mutations than is in fact observed.  What is the linguistic equivalent of lactose intolerance?

(Click here for our satire on the subject, which led to this whole U.M.T. thread in the first place.)  We might leave it open, just where these structures do reside:  perhaps upon that same hillside where the qualities of being Abelian, distributive, semi-simple, etc., may be found.