A New Characterization of the Continuous Functions on a Locale |
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Authors: | Richard N. Ball Anthony W. Hager |
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Affiliation: | (1) Department of Mathematics, University of Denver, Denver, CO 80208, USA;(2) Department of Mathematics, Wesleyan University, Middletown, CT 06459, USA |
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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 c3A, the ``closed under countable composition hull of A,' as C(YlA), where YlA is the Yosida locale of A, and as the largest essential reflection of A. |
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Keywords: | Primary: 06F20 06F25 Secondary 46E05 46E25 54E15 |
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