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Localic completion of generalized metric spaces I

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Steven Vickers

Following Lawvere, a generalized metric space (gms) is a set $X$ equipped
with a metric map from $X^{2}$ to the interval of upper reals
(approximated from above but not from below) from 0 to $\infty$
inclusive, and satisfying the zero self-distance law and the triangle
inequality.

We describe a completion of gms's by Cauchy filters of formal balls. In
terms of Lawvere's approach using categories enriched over $[0,\infty]$,
the Cauchy filters are equivalent to flat left modules.

The completion generalizes the usual one for metric spaces. For
quasimetrics it is equivalent to the Yoneda completion in its netwise form
due to Kunzi and Schellekens and thereby gives a new and explicit
characterization of the points of the Yoneda completion.

Non-expansive functions between gms's lift to continuous maps between the
completions.

Various examples and constructions are given, including finite products.

The completion is easily adapted to produce a locale, and that part of the
work is constructively valid. The exposition illustrates the use of
geometric logic to enable point-based reasoning for locales.

Keywords:
topology, locale, geometric logic, metric,
quasimetric, completion, enriched category

2000 MSC:
primary 54E50; secondary 26E40, 06D22, 18D20, 03G30

*Theory and Applications of Categories,*
Vol. 14, 2005,
No. 15, pp 328-356.

http://www.tac.mta.ca/tac/volumes/14/15/14-15.dvi

http://www.tac.mta.ca/tac/volumes/14/15/14-15.ps

http://www.tac.mta.ca/tac/volumes/14/15/14-15.pdf

ftp://ftp.tac.mta.ca/pub/tac/html/volumes/14/15/14-15.dvi

ftp://ftp.tac.mta.ca/pub/tac/html/volumes/14/15/14-15.ps

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