Geodesic
Non-Metrizable Uniformities and Proximities on Metrizable Spaces
Canadian journal of mathematics, Tome 25 (1973) no. 5, pp. 979-981

Voir la notice de l'article provenant de la source Cambridge University Press

In the literature there exist examples of metrizable spaces admitting nonmetrizable uniformities (e.g., see [3, Problem C, p. 204]). In this paper, this phenomenon is presented more coherently by showing that every non-compact metrizable space admits at least one non-metrizable proximity and uncountably many non-metrizable uniformities. It is also proved that the finest compatible uniformity (proximity) on a non-compact non-semidiscrete space is always non-metrizable. 
Sharma, P. L. Non-Metrizable Uniformities and Proximities on Metrizable Spaces. Canadian journal of mathematics, Tome 25 (1973) no. 5, pp. 979-981. doi: 10.4153/CJM-1973-103-7
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[1] 1. Alfsen, E. M. and O. Njåstad, Proximity and generalized uniformity, Fund. Math. 52 (1963), 253–252. Google Scholar

[2] 2. Gillman, L. and Jerison, M., Rings of continuous functions (Van Nostrand, Princeton, 1960). Google Scholar

[3] 3. Kelley, J. L., General topology (Van Nostrand, Princeton, 1961). Google Scholar

[4] 4. Naimpally, S. A. and Warrack, B. D., Proximity Spaces, Cambridge Tract in Maths., No. 59 (Cambridge University Press, Cambridge, 1970). Google Scholar

[5] 5. Sharma, P. L., Compactification numbers (preprint). Google Scholar

[6] 6. Sharma, P. L., On p-classes of uniformities (preprint). Google Scholar

[7] 7. Reed, E. E. and Thron, W. J., m-bounded uniformities between two given uniformities, Trans. Amer. Math. Soc. 141 (1969), 71–77. Google Scholar

[8] 8. Thron, W. J., Topological structures, (Holt, Rinehart and Winston, New York, 1966). Google Scholar

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