A New Characterization of the Continuous Functions on a Locale

A New Characterization of the Continuous Functions on a Locale Within the category W of archimedean lattice-ordered groups with weak order unit, we show that the objects of the form C(L), the set of continuous real-valued functions on a locale L, are precisely those which are divisible and complete with respect to a variant of uniform convergence, here termed indicated uniform convergence. We construct the corresponding completion of a W-object A purely algebraically in terms of Cauchy sequences. This completion can be variously described as c 3 A, the ``closed under countable composition hull of A,'' as C(Y l A), where Y l A is the Yosida locale of A, and as the largest essential reflection of A. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

A New Characterization of the Continuous Functions on a Locale

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Publisher
Birkhäuser-Verlag
Copyright
Copyright © 2006 by Birkhäuser Verlag, Basel
Subject
Mathematics; Fourier Analysis; Operator Theory; Potential Theory; Calculus of Variations and Optimal Control; Optimization; Econometrics
ISSN
1385-1292
eISSN
1572-9281
D.O.I.
10.1007/s11117-005-5826-5
Publisher site
See Article on Publisher Site

Abstract

Within the category W of archimedean lattice-ordered groups with weak order unit, we show that the objects of the form C(L), the set of continuous real-valued functions on a locale L, are precisely those which are divisible and complete with respect to a variant of uniform convergence, here termed indicated uniform convergence. We construct the corresponding completion of a W-object A purely algebraically in terms of Cauchy sequences. This completion can be variously described as c 3 A, the ``closed under countable composition hull of A,'' as C(Y l A), where Y l A is the Yosida locale of A, and as the largest essential reflection of A.

Journal

PositivitySpringer Journals

Published: Jan 1, 2005

References

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