Re: -- The "most natural" completion of a metric space

On Mon, 06 Jul 2009 02:29:43 EDT, omega <blue_jaguar@xxxxxxxxx> wrote:

i'm not up to speed on category theory notation, so
forgive me if what i'm saying is trivial to you, but

the completion of any metric space is unique up to
isometry,

Very true. And this is the "completion functor" between the skeletons [Met] and [CMet], i.e. when
any two metric spaces are identified if they are isometric.

so looking for a "most natural" completion
is not necessary

My point is the completion of a given metric space is in fact a class of isometric but different
"concrete relizations", and the problem is : how to *choose* among them still in a *functorial* way,
without using any nontrivial equivalence.

E.g., if N is a normed space, one can view its completion either as the closure of N into N**, or
as the closure of N into C(S), where S is the closed unit ball of N*, endowed with the w*-topology.
Or in many other (equivalent, but distinct) ways (e.g., using the construction with Cauchy sequences).
Which is "the most adequate" to be nominated as *the* completion of N ?

So that, I'm looking for a completion functor that preserves the "concrete nature of the objects", i.e.,
not identifying two different metric spaces, even that they are isometric. And also preserving the
complete metric spaces. (I do not know if such a "lifting" is possible.)

I'm really not sure what the requirement is, but have you condsidered
the standard construction of the completion of a metric space?

Say X is a metric space. Let C be the space of Cauchy sequences of
elements of X. Say two sequences (x_n), (y_n) in C are equivalent
if d(x_n, y_n) -> 0. Then the quotient of C by this equivalence
relation is the completion of X (with the metric defined in the
obvious way and with the obvious embedding of X; both
those are left as exercises if they're not obvious...)


David C. Ullrich

"Understanding Godel isn't about following his formal proof.
That would make a mockery of everything Godel was up to."
(John Jones, "My talk about Godel to the post-grads."
in sci.logic.)
.

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