Geodesic
On topological spaces with a unique compatible quasi-uniformity
Glasgow mathematical journal, Tome 18 (1977) no. 1, pp. 11-12

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It is shown in [2] that a uqu space satisfies the following conditions.(DC) There is no infinite, strictly decreasing sequence of open sets with open intersection.(IC) There is no infinite, strictly increasing sequence of open sets.In this note we show that for a transitive space these conditions are sufficient for the space to be uqu. This will follow as a consequence of the following result. 
Brown, Lawrence M. On topological spaces with a unique compatible quasi-uniformity. Glasgow mathematical journal, Tome 18 (1977) no. 1, pp. 11-12. doi: 10.1017/S0017089500002962
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[1] 1.Birkhoff, G., Lattice theory, 3rd ed., Amer. Math. Soc. Colloquium Publications 25 (1967). Google Scholar

[2] 2.Lindgren, W. F., Topological spaces with a unique compatible quasi-uniformity, Canad. Math. Bull. 14 (1971), 369–372. Google Scholar | DOI

[3] 3.Szasz, G., Einführung in die Verbandtheorie (Teubner, Leipzig, 1962). Google Scholar

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