Vol. 43, No. 1, 1972

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The nonstandard hulls of a uniform space

C. Ward Henson

Vol. 43 (1972), No. 1, 115–137

Let (X,𝒰) be a uniform space in some set theoretical structure and let x be the set corresponding to X in an enlargement ∗ℳ of In this paper a set of 𝒰-finite elements of X is defined and this set is used to define a nonstandard hull of (X,𝒰). The main result is that, with some specific exceptions depending on the existence of measurable cardinal numbers, this nonstandard hull is the same as the smallest of the nonstandard hulls defined by Luxemburg. This result is used in giving a characterization of subsets of X on which every uniformly continuous, real valued function is bounded. Also, two examples are given to illustrate the possible structure of the nonstandard hulls.

Mathematical Subject Classification 2000
Primary: 54E15
Secondary: 02H25
Received: 30 July 1971
Published: 1 October 1972
C. Ward Henson