Background:
Topology is familiarly, informally characterized as “rubber-sheet
geometry”. That is, unlike the
Euclidean geometry that we learned in school, which applies to flat rigid
surfaces, you’re allowed to
stretch and bend the space, so long as you don’t tear it or let it intersect
itself.
But at some point, we might like -- without returning quite
to the simplicities and rigidities of the Euclidean picture -- to make our
space… a bit less rubbery.
As the godfather of Calabi-Yau manifolds puts it:
We start with some raw topological
space, which is like a bare patch of land that’s been razed for
construction. On top of that, we’d
like to build some kind of geometric structure that can later be decorated in
various ways.
-- Shing-Tung Yau, The Shape of
Inner Space (2010), p. 77
[Note from Jan 2011]
We were nonplussed to learn that this site comes up on the first page of Google search on “Urysohn Metrization Theorem”.
[Note: This has since changed, owing to a hack attack by the Nominalist Internationale.]
[Metanote: It's back again, thanks to a counteroffensive by the Realist Underground.]
And ill at ease, since
our post of that title is a satire on sociobiological/ultraDarwinistic overreach, a satire of a sort practiced almost a century ago by G.K. Chesterton in his book
The Everlasting Man. Pity the unsuspecting physicist or math major who winds up
there in hopes of learning the first thing about metrization, Urysohn or otherwise. So we feel we owe it to these blameless Internauts to offer them at least a little something for their trouble. Here, then, for the non-mathematician, or (God willing) the mathematician-to-be, is a thumbnail sketch of what led to this theorem in the first place.
The intuitive content of the theorem is as follows. If you have a space with enough structure to keep things apart which ought to be, and if the space itself is not too huge, then you can define a distance between any pair of elements. The function that specifies this distance is called the “metric”, from the Greek word for 'measure'.
Thus, you mightn’t be able to do this if you lived in an oozy sort of world, where the minimal entities were like blobs with sometimes inextricably intertwined tentacles; nor if your world were scattered among separate universes.
(Footnote: the idea is that you can come up with a nontrivial metric. After all, any set whatever can be regarded as a (trivial) metric space, given the discrete topology.)
As for formal statements, these vary somewhat. Here is a sampling. (The following assemblage is an atavism from my days as a lexicographer at Merriam-Webster; we worked from piles of attestation-slips, called "cites".)
John Kelley, General Topology (1955), p. 125:
Metrization Theorem (Urysohn)
A regular T1-space whose topology has a countable basis is homeomorphic to a subspace of the [Hilbert] cube and hence metrizable.
James Dugundji, Topology (1965), p. 195, formulates it as
In 2-countable spaces, regularity is equivalent to metrizability.
and he labels this merely a “corollary” of
Theorem (Nagata and Smirnov) A topological space is metrizable if and only if it is regular and has a basis that can be decomposed into an at most countable collection of neighborhood-finite families.
The same can be said for the Bing metrization theorem, which likewise sharpens the sufficient condition into one both sufficient and necessary: “a topological space X is metrizable if and only if it is regular and T0 and has a σ-discrete basis.” (Wiki)
Other formulations:
Stephen Willard, General Topology (1970), p. 166:
Urysohn’s metrization theorem. The following are equivalent for a T1-space X:
(a) X is regular and second countable
(b) X is separable and metrizable
(c ) X can be embedded as a subspace of the Hilbert cube.
James Munkres, Topology: a First Course (1975), p. 217:
Urysohn’s metrization theorem. Every regular space with a countable basis is metrizable.
Michael Henle, A Combinatorial Introduction to Topology (1979), p. 283:
Metrization Theorem (Urysohn)
A compact Hausdorff space that is second countable is a metric space.
That one uses a stronger condition to reach the same conclusion, and is thus a weaker theorem; the same version appears here:
Boto von Querenburg, Mengentheoretische Topologie (3rd edn. 2001):
Ein kompakter Hausdorff-Raum is genau dann metrisierbar, wenn er eine abzählbare Basis besitzt.
Something of an odd-man-out, possibly importing the stronger “normality” condition from the Urysohn Lemma, is this:
Seymour Lipschutz, General Topology (1965), p. 142:
Urysohn’s metrization theorem. Every second-countable normal T1-space is metrizable.
But cf. this:
George Simmons, Introduction to Topology and Modern Analysis (1963), p. 138, which offers a slightly stronger version, and names it differently:
Urysohn Imbedding Theorem. If X is a second-countable normal space, then there exists a homeomorphism of X onto a subspace of R-to-the-infinity, and X is therefore metrizable.
And indeed, the Lipschutz formulation is echoed much more recently in the October 2010 American Mathematical Monthly (“A Tale of Topology”, by Gerald Folland):
Every second-countable normal space is metrizable.
If all that already makes sense to you and seems obvious, you’re done. If not, read on.
*
The first order of business is to motivate the theorem. What does it mean for a space to be metrizable, and why should we care?
The space we’re best familiar with is the one we live in; but the one we have studied most analytically, traditionally in high school geometry class, is the nice flat one, called the Euclidean plane. This we studied first by Euclid’s own methods, which date back over two thousand years, with axioms and proofs that justify each step -- the best possible mental exercise -- and lots of diagrams. Later (if we stay the course) we take up a new approach, using analytic methods, which largely began with Descartes, in the seventeenth century. Here we add a grid of axes, which measures exactly where each point is and how far apart they are, and prove things about figures: now not just triangles and circles and rectangles, but hyperbolas and cycloids and any shape we want, by means of equations. You don't have much in the way of equations with Euclid; for that, you need numbers -- given by the metric.
And lo -- already, in these simple memories of high school, we have, in miniature, a picture of what has happened at the forefront of mathematical research over the past century or so. For geometry, in the sense with which you are all familiar, came to be generalized to a new subject, topology (originally called analysis situs -- both mean ‘the study of place’, as geometry means ‘the measuring of the earth’). Whereas the Euclidean plane is rigid, we let these spaces get all stretchy and bendy. In that case we can no longer say what the circumference of a circle is, because by the time we wake up in the morning it may have stretched and drooped like one of Salvador Dali’s watches (in his painting, “The Persistence of Memory”): but some things do remain true, such as the fact that that curve has an inside and an outside, meaning you can’t get there from here without crossing that curve. (Note: Such entirely general, almost naively simple-sounding statements are typical of topology. The content of that one is called the Jordan Curve Theorem, and it's a real bear to prove.)
Now topology was originally point-set topology, which mentally is rather like Euclid’s geometry: you set up the ground rules for a space, then you ponder and visualize and reason things through, using pictures if you possibly can, and your own intuition. Meanwhile, behind the scenes, a new view of topology was taking shape, somewhat analogous to what Descartes did for (or to) geometry: instead of reasoning, half-intuitively, with spaces and shapes, you come up with an algrebra whose structures manage to reflect what is going on in those spaces in more detail, yielding numbers and equations and things you can calculate with. It could have been called “analytic topology” by analogy with “analytic geometry”, but instead it is called algebraic topology. Though very powerful, it is somewhat bloodless (at least for the beginner), and requires different habits of mind. (Habits I alas lack. Readers of my tales of woe will recall my bruising encounter with that subject; the spot still smarts in frosty weather yet.)
So: Cartesian geometry takes us, from shapes, to the antecedently familiar realm of equations involving numbers. Homology (a part of algebraic topology) takes us from more general shapes to the relatively tractable algebraic structures called Abelian groups.
The distance function in Cartesian geometry is what you get from the Pythagorean theorem. The criteria for the general topological notion of a metric are a straightforward abstraction from this: mainly, if you make a beeline from here to there, and another beeline from there to yonder, the distance traveled must be at least as much as had you simply gone straight to yonder from here. As to what-all can meet the criteria -- ah, there lie surprises.
Euclidean geometry is described as what you can do with a straight-edge and compass. Sometimes people say “ruler” and compass, but that is a mistake: we have no measurement-markings on our straight-edge; there are no pre-established
units of measurement. We can still determine that two different line-segments are the
same length -- just take our trusty compass, measure the first segment with it, and now see if that compass-setting matches the endpoints of the second segment. You might say that, in this world,
length itself is not absolutely defined, whereas
being-as-long-as is. (This observation could be pursued in a syntactic direction -- that of
incomplete symbols -- with interesting results.)
Now, in the Cartesian approach -- analytic geometry -- we want to work with actual numbers, because that speeds things up. So we turn the plane into a metric space -- “metric” just means ‘measurement’. And the reason it is possible to do so is that the (pre-Cartesian) Euclidean plane was already rigid: you do not change the length of something simply by moving it about; you can slide one triangle over to another one and see if they’re congruent.
Furthermore, the way we shall conveniently measure things was already suggested to us by the celebrated truth of Euclidean geometry, expressed in the Pythagorean Theorem. The earlier formulation of this was: “the square on the hypotenuse is equal to the sum of the squares on the other two sides”: meaning, the area of a square figure, one of whose sides is the long side (the ‘diagonal’) of a right-angled triangle, is equal to the sum of the areas of two other triangles likewise sticking off the shorter sides that are perpendicular to each other. This was still a ‘point-set’ geometric view. But now we start writing it in symbols, saying that, if x is the length of the one leg, and y is the length of the other, then the length of the diagonal is the square root of the sum of x-squared plus y-squared. (This is known as the “Euclidean metric”.) That’s algebra. And the new viewpoint is reflected in the way you’ll here the theorem quoted nowadays: “the square of the hypotenuse is equal to the sum of the squares of the other two sides”:
The rest you are familiar with. We briskly mark off a bunch of equal lengths along one direction, which we call the x-axis, and likewise along the y-axis that is perpendicular to it, and from this we get graph-paper, with its familiar grid. And now our old friend the plane, which we first came to know as a tabula rasa -- the plane itself, and our own childish minds -- wears its Metric Space status on its sleeve, so to speak.
This is all so familiar, that we are in danger of letting our memories do the thinking for us. For in fact there are many different ways of deciding to set of a scheme of measurement on a flat surface. We might stipulate that the ‘distance’ between two points shall be simply whichever is
larger, the difference in the x-value or the difference in the y-value. Or we might say instead that the distance shall be the square root of the
difference of x-squared and y-squared, rather than their sum. This is what Minkowski did, and it turns out to be the key to uniting space and time into a single Metric Space -- spacetime. These and others are alternative possible
metrizations of a plane. (In one of them, it is possible to draw a
round square -- the paradigm example of what philosophers tell us is impossible. You can read about one
here.)
Of course, we don’t ourselves live inside of piece of paper (as Flatlanders do), we live in nice big rooms -- three dimensions rather than two. We set up a third axis, the z-axis, like a tent-pole, to give us some breathing-space: and now the grid shapes are little cubes instead of little squares. Using this Euclidean metric, it turns out that an analog of the Pythagorean Theorem holds here as well: we can consistently define the distance as the square root of the sum of x-squared plus y-squared plus z-squared. Analytic geometry proceeds as before, with barely any change in methods (I originally wrote, "since the ones we used in the two-dimensional case were already so powerful"; but actually that puts the cart before the horse: it is precisely such (unexpected) generalizability of a method that leads us to call that method 'powerful'.). So now, instead of just circles and parabolas and so forth, we have a richer world of shapes, like cones and spheres and ellipsoids and helices, and on and on. (Actually these were already known to the Greeks, though how they managed it with the pre-Cartesian methods they had, is something of a miracle.) And it continues to be easy to prove things about these, since we still have basically the same metric, which is well adapted to equations and their numerical solutions. Likewise in four dimensions, and on up as high as you like.
(Note: People get all spooky when they hear things like 'fourth dimension', but these metric methods absolutely tame them. -- Children: Study math.)
Bottom line: A metric space is a very convenient thing to work with. You can pretty much know where you are and do what you want, even when the space gets hairy in other ways, like being infinite-dimensional, or very curvy.
But.
In the inexhaustible splendor of the Creation, there are many many different spaces, more numerous than the stars. They sprang full-blown from the Creator’s brow, and it is up to us to discover their structure. Unfortunately, when we first meet up with one of these, it may not be wearing a nice convenient metric on its sleeve. It may be a very confusing, huge, menacing, squishy blob. -- Recall that when we first met the plane, it too came without a pre-drawn grid. But a particular grid was already implicit, because Euclid’s axioms imply the Pythagorean theorem. (There are other metrics you could adopt where that theorem wouldn’t be true, but these would not conform to our everyday local experience, which is why Euclid chose the one he did, and why it took two millennia to generalize the naive notion of "distance" to the mathematical notion of "metric".)
So: Faced with such a blob, can we come up with a consistent measuring-scheme that will tame it, by turning it into a metric space? That is, is it metrizable? -- And the answer is: Sometimes you can, and sometimes you can’t. So, we want to come up with ways we can tell, whether the project is doable or hopeless. It’s a bit like figuring out whether you can tame a given kind of animal. Long ago, people figured out that you could do that with dogs, and later with horses, to a huge extent, so that instead of being wild beasts they are actually useful. Cats, it turns out can be tamed to a lesser extent -- tamed to tolerate us, so long as we are not late with the cheeseburgers. They’re not useful, but they’re decorative. (Meanwhile in Catland, the lecture reads: “Peeps are redonkully e-z 2 tame; goggies, not so much.”) So, you see a puppy and, no matter what breed it is, the mere fact that it is Canis familiaris tells you that you have an excellent chance of taming it: though just how to do so may vary with the breed. In a similar fashion, when we first meet an untamed space at the Space Store, we can know, by certain signs (which Urysohn specifies in his theorem) that the thing is in principle metrizable -- though it doesn’t give us a useful metric just for free; for that we still have to do some work.
*
If you and I are points in a metric space, the metric tells us how far apart we are: that’s like analytic geometry. But topology lets out the sails a bit. In the most general topological space, you can’t say how far apart two points are -- but you can always say what bunks with what. The bunks are known as “neighborhoods” or (roughly synonymous) “open sets”, and are given as part of the very definition of the space. Thus, the most featureless space of all, justly (in this metaphor) called the “indiscrete” space, everybody bunks with everyone else, no privacy at all. In the opposite, the “discrete” space (discrete, not discreet), each man is an island. But most spaces, and every space of interest, is in between.
To describe just where they fall on this in-between spectrum, we name various “separation” properties. The simplest common one is that any two points can be separated (any two people can sleep in separate bunks). In a metric space, this is easy. If you and I are a certain distance apart, then if I draw (if we’re in a plane) a circle or (if we’re in a fatter space) a sphere around me, with a radius less than that distance, then I’m in my own special bubble and you’re outside it. Such elementary capacity of separation is also available in most non-metric spaces. This basic degree of separation is called T1. If you and I can each draw such a neighborhood simultaneously, even better -- the space is “Hausdorf”.
A more demanding requirement is that I can fix around myself a bubble (a neighborhood) which keeps me clear, not of just a single point, but of any collection of points called a “closed” set. A set is closed if, for any point you can creep up on, in an infinite sequence, that point is in the set. So for instance, on the number-line, take all the reciprocals of the natural numbers, ½, 1/3, ¼, etc: these creep up on zero -- they get as close as ever you please, so the set of these reciprocals is not closed: to close this set, you have to add zero. Or, take everything inside a circle. You can creep up to any point on the boundary, from within the circle, so the interior is not closed. Add the boundary, now it’s closed. -- So: given a point, can that point stay clear (hide inside a bubble), not just from any other point (that’s easy), but from any other closed set that doesn’t contain that point -- no matter how pushy and encroaching? Well, if that set is closed, it can’t keep creeping up on me indefinitely: at some stage, it can’t come any closer, otherwise I’d be a limit point of that set, and since I’m not in that set, it wouldn’t be closed. So at some point it keeps its distance -- I’m safe in my neighborhood (say, Beacon Hill), where the menacing set cannot encroach. (In a metric space, this is easier to visualize: it keeps its distance -- say, d. So I draw a little bubble round me, of radius smaller than d, and I’m safe.( -- Spaces that are like this are called regular. Obviously, every regular space is Hausdorf, but not vice versa.
(The next step up is: Can any two closed sets -- not just one closed set and a point -- be kept apart by disjoint neighborhoods (neighborhoods that don’t intersect)? If so, that space is called normal. Normality is used in the Urysohn lemma, basically unrelated to the UMT.)
So, being regular suffices to keep things separate enough to define distances between things. But regularity by itself is not sufficient -- the space might be too ‘big’ to fit in a metric. How big is too big? Bigger than second-countable, the other premise of the U.M.T. In that case, points can be just too far apart to have a finite distance.
*
None of these definitional and formal considerations
gets across the real power of the metric-space idea.
This arises when we begin to consider a more abstract sort of space, in which the “points” are not characterless, dimensionless ideal dots, but …
functions.
(This is an example of the “
Ladder ofAbstraction”.)
Andrew Gleason puts the matter well:
The assignment of a metric to a set of functions gives this set an intuitively geometric character. The success of the theory of metric spaces in analysis can be attributed to the remarkable insight into the nature of functions which has come from exploiting the geometric point of view.
-- Andrew Gleason, Fundamentals of Abstract Analysis (1966), p. 226
