Saturday, January 15, 2011

Footnote to "The Urysohn Metrization Theorem (continued)"

It turns out they can still be quite wild.  For even among the metrizables, we may find that we have bitten off  more than we can comfortably chew.  Cf. Lynn Loomis and Shlomo Sternberg, Advanced Calculus (1968):

A metric space can be a very odd object.  It may fail to have almost any property one can think of.

Apart from regularity and second-countability, I was about to add;  and yet, reviewing the theorem-statements quoted earlier, not a one states clearly that these two qualities are both sufficient and necessary for metrizability.  The closest we come is with Willard:

The following are equivalent for a T1-space X:
(a)  X is regular and second countable
(b) X is separable and metrizable

which says only that if the space is both metrizable and separable, then the other two qualities follow.
            So:  a fortiori:  having taken what we might have imagined a fairly innocuous step of expanding the field, we are overwhelmed by the abundant results, and quickly seek further constraints to make matters manageable.   There are stronger versions of metrics, such as the norm (homogeneous, translation-invariant metrics) and (stronger still) the inner product, trademark of Hilbert Space. 
            The situation recalls that of post-Chomskyan linguistics, where the giddy sense of unlimited possibilities that came with a naïve conception of generative grammar, soon gave way to a sober search for constraints, lest the theory be devoid of interesting theorems.

[Concluded here.]

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