The Urysohn Metrization Theorem (continued)

[continuing this]

 "Once you know a metric exists, there are all kinds of consequences.  You can use that knowledge to work backward  and deduce things about manifold topology, without knowing the exact metric."
-- Shing-Tung Yau, The Shape of Inner Space (2010), p.123


You have all seen topology defined as “rubber-sheet geometry”, and watched the proverbial coffee-cup  morph into its natural complement, the donut.  For such general spaces, distance between two points is not automatic, since you can stretch the points apart.   Still, there is plenty to talk about, in purely topological and non-metric terms: questions of connectivity, compactness, dimension, and the fineness of mesh of the neighborhood system.   If we are able to impose a metric, we get new topics, like (radius of) curvature.

The U.M.T. nicely illustrates what we might call the Dialectic of the Topological Enterprise.

Thesis:  Traditional Euclidean Geometry.

Antithesis:  Extract the essentials, and open up the club to anything matching that.

Synthesis:  Yipes!  There are more spaces in heaven and earth, than were dreamt of in our philosophy.   Impose additional conditions to make some of them manageable.

Since a metric space allows you to do all sorts of traditional and well-understood operations, it is especially pleasurable to discover what initially might look like quite wild spaces, which nevertheless are metrizable. (Typically, these are spaces in which a "point" is a whole function, and not a 'dot'.)

And having taken that step, there are new antitheses, such as expanding the notion of “metric” to that of a pseudo-metric (which, however, has not been nearly so rich a generalization, so far as I know).  Here you let nondistinct points have zero distance between them.  Such, in effect, is what we do in our actual cosmos, when, in calculating a distance between two points in classical four-dimensional spacetime, we ignore the tiny compactified extra  dimensions which (in Kaluza-Klein or String Theory)  curl around each one of these points.   -- A more fruitful extension of Euclidean distance, and quite unexpected, has been “p-adic distance”, which would get us into very deep waters indeed.

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A quite different  and likewise lasting  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 !

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