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Compactness in apartness spaces? Bridges, Douglas Ishihara, Hajime Schuster, Peter Vita, Luminita S. Apartness constructive compact uniform space A major problem in the constructive theory of apartness spaces is that of finding a good notion of compactness. Such a notion should (i) reduce to ``complete plus totally bounded'' for uniform spaces and (ii) classically be equivalent to the usual Heine-Borel-Lebesgue property for the apartness topology. The constructive counterpart of the smallest uniform structure compatible with a given apartness, while not constructively a uniform structure, offers a possible solution to the compactness-definition problem. That counterpart turns out to be interesting in its own right, and reveals some additional properties of an apartness that may have uses elsewhere in the theory. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Douglas Bridges and Hajime Ishihara and Peter Schuster and Luminita S. Vita 2005 Is Part Of Dagstuhl Seminar Proceedings, Volume 4351, Spatial Representation: Discrete vs. Continuous Computational Models (2005) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/DagSemProc.04351.9 urn:nbn:de:0030-drops-1175 https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.04351.9 eng https://creativecommons.org/licenses/by/4.0/legalcode