Geodesic
On Metric Completeness and Order Completeness
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Geometric topology and set theory, Tome 247 (2004), pp. 228-236
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A positive answer to the question of whether there exists a metric on the lattice of continuous functions that generates uniform convergence and is such that the metric completion is simultaneously the order completion is given. Two interpretations of the obtained metric lattice are suggested. 
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S. N. Samborskii. On Metric Completeness and Order Completeness. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Geometric topology and set theory, Tome 247 (2004), pp. 228-236. http://geodesic.mathdoc.fr/item/TM_2004_247_a15/

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