# On Uniform Homeomorphism between Two Uniform Spaces

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## Abstract

Journal of the Institute of Polytechnics, Osaka City U niversity, Vol. 3, No, 1 -2 , Series A On Uniform Homeomorphism between Two Uniform Spaces By Jun-iti Nagata (Received September 15, 1952) The purpose of this paper is to study conditions in order that two complete uniform spaces are uniformly homeomorphic. We concerned ourselves with the same problem in a previously published paper.i^ There we characterized a point by a family of uniform coverings, and for that the condition of the lattice of uniform basis as well as the proofs of theorems was considerably complicated and unnatural. By using a family of families of uniform coverings in the place of a family of uniform coverings, in this paper we show that the condition I ) in the previous paper and a condition, weaker than 2) there are sufficient for the conditions of the lattice of uniform basis which defines the uniform complete space up to a uniform homeomorphism, and we simplify proofs of propositions. We concern ourselves with the lattice L(^R) of uniform basis of a complete uniform space EP , satisfying the following conditions, 1) i f ^U R ) , then ^LQR). 2) i f then for an arbitrary open set Z7o» there exists ^KC/o»U) in LCR) such that i) M€DJl(C7o»U) implies M ^ U o* ii) u e v i and Ur^Uo=<t> imply C/CM 53) fo r some M. 1) is the same condition as I ) in the previous paper. 2) is weaker than 2 ') in the previous paper and accordingly than 2) there. Definition. We denote by [iAiP) the set of all the families jul of uniform coverings such that for every nbd ( —neighbourhood) U ip ) of p, there exists 50^6;« for which U(P)CtM for all M €931. Lemma I. \fj](<P) satisfies the following two conditions I) fo r every U € £ (i? ) , there exist W ^ L (R ) and % ^ L (R ) such that U'<H, ; U '<^v5x and imply U < $ v O ; ^9)1 fo r every /i and some 9)1 G 1) On conditions in order that two uniform spaces are uniformly homeomorphic, this journal. Vol. 2, No. 2, 1952. 2) L{R) is a fa

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