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Uniform Structures Compatible with a Given Topological Space

Dr. S. M. Padhye , Priti. P. Umarkar

Given a topological space (X,𝒯), we construct uniformities on X compatible with topology of X. In particular it is proved that the uniformity 𝒰 generated by continuous real valued functions on X is compatible with topology of X if and only if X is completely regular. Further it is also proved that the uniformity 𝒰1 generated by continuous bounded real valued functions on X is also compatible with topology of X under the same hypothesis. In many cases all the uniformities are different. In particular all these uniformities coincide for a compact Hausdorff space.