Geodesic
Uniqueness of Compatible Quasi-Uniformities
Canadian mathematical bulletin, Tome 15 (1972) no. 4, pp. 575-583

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It is shown that a topological space has a unique compatible quasi-uniformity if its topology is finite. Examples are given to show the converse is false for T 1 and for normal second countable spaces. Two sufficient conditions are given for a topological space to have a compatible quasi-uniformity strictly finer than the associated Császár-Pervin quasi-uniformity. These conditions are used to show that a Hausdorff, semi-regular or first countable T 1 space has a unique compatible quasi-uniformity if and only if its topology is finite. Császár and Pervin described, in quite different ways, quasi-uniformities which induce a given topology. It is shown that, for a given topological space, Császár and Pervin described the same quasi-uniformity. 
Votaw, Charles I. Uniqueness of Compatible Quasi-Uniformities. Canadian mathematical bulletin, Tome 15 (1972) no. 4, pp. 575-583. doi: 10.4153/CMB-1972-100-9
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     title = {Uniqueness of {Compatible} {Quasi-Uniformities}},
     journal = {Canadian mathematical bulletin},
     pages = {575--583},
     year = {1972},
     volume = {15},
     number = {4},
     doi = {10.4153/CMB-1972-100-9},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-1972-100-9/}
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