Subcategories of uniform spaces

Rice, Michael D.

doi:10.1090/S0002-9947-1975-0358708-2

Subcategories of uniform spaces
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Trans. Amer. Math. Soc. 201 (1975), 305-314 Request permission

Abstract:

The problem of embedding a topological space as a closed subspace of a product of members from a given family has received considerable attention in the past twenty years, while the corresponding problem in uniform spaces has been largely ignored. In this paper we initiate the study of the closed uniform subspaces of products of metric spaces. In §1 we introduce the functor $m$, which is used in §2 to characterize the closed subspaces of products of metric spaces and separable metric spaces, and the closed subspaces of powers of the open unit interval $(0, 1)$. In §3 we obtain various descriptions of the functor $d$ which associates to each uniform space a closed subspace of a product of metric spaces and establish the equation $md = dm$. This leads to a characterzation of the completeness of $euX$, the uniform space generated by the countable $u$-uniform covers, in terms of the completeness of $uX$ and a countable intersection property on Cauchy filters.

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