Monday, July 4, 2016

The Urysohn Metrization Theorem: a Perceptualist approach

At least since Locke, and down to our own day, an especially gormless form of empiricism, which we may call perceptualism, has held sway.
This theory is too boring even to summarize, let alone polemicize against, let alone analyze;  but -- credit where due -- we offer a

Perceptualist Proof of the UMT

(1)  I see … an array of color-patches, in my visual field. …  They form  (or so it seems to me, at this particular instant) … a topological space.

(2) They -- the pattern -- I see at once (unless I am but dreaming) -- it’s:  Regular !!

(3)  I focus on a single point (“Andrew”), and the topology there.  It has, one imagines, were there world enough and time to actually count and Booleanize all the open sets (1, 2, 3, …. ) a countable basis.
I try another point (“Bertha”):  likewise, the basis of the topology there is (let us say  -- obviously we could never really know this, any more than we can truly know that we have hands) countable.
Might this be the general case ??

(4) And -- yes! -- Charlie and Denise and Edmund and … all are graced with a topology enjoying a countable basis!   The space as a whole thus has one [NDLR: Fallacious step, but what do you expect] -- it’s Second-Countable !!!

(5)  And now … hoving in from beyond the horizon, as though from nowhere, it’s … look … a metric!  A very metric!  Hurrah!  The space is metrizable !!!!!!!

(6)  Q.E.D.

No comments:

Post a Comment