Geodesic
On Metric Completeness and Order Completeness
Informatics and Automation, Geometric topology and set theory, Tome 247 (2004), pp. 228-236.

Voir la notice de l'article provenant de la source Math-Net.Ru

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. Informatics and Automation, Geometric topology and set theory, Tome 247 (2004), pp. 228-236. http://geodesic.mathdoc.fr/item/TRSPY_2004_247_a15/

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[3] Samborskii S. N., “O rasshireniyakh differentsialnykh operatorov i negladkikh resheniyakh differentsialnykh uravnenii”, Kibernetika i sistemnyi analiz, 2002, no. 3, 163–180 | MR | Zbl