Studia Scientiarium Mathematicarum Hungarica 29. (1994)

1-2. szám - Deák J.: On proximally fine contiguities

Studia Scientiarum Mathematicarum Hungarica 29 (1994), ON PROXIMALLY FINE CONTIGUITIES J. DEÁK Abstract We investigate some properties of the finest contiguity compatible with a proximity (in the sense of őech). Counterexamples show that most of the analogous statements are false for Riesz or Lodato proximities. The paper ends up with an open problem. Given a proximity 6 in the sense of Cech [1] (respectively a Riesz or Lodato proximity), let T1(<5) (respectively 1^(6) or r^(<5)) denote the finest contiguity (Riesz or Lodato contiguity) compatible with S; we aim at inves­tigating the properties of the functors T1, and The two statements labelled “Theorem” answer questions raised by Prof. Á. Császár. § 0. Preliminaries Let us recall some definitions and simple facts (see e.g. [2] § 0, the refer­ences given in [2], and [3] § 6). 0.1 A symmetric relation 6 between subsets of X is a proximity on X if (i) A S X implies (ii) A fl B ^ 0 implies AS B; (iii) A 6 (B U C) iff either AS B or A S C. The proximity 6 is Riesz if AS B implies c(A) fl c(B) = 0, where S means that S does not hold, and c(A) = {x: {x} S A}; it is Lodato if ASB implies c(A)Sc(B). If S is Lodato then c is the closure in a topology, which is called the topology induced by S. A filter f on X is S-compressed if ASB whenever A, B £ secf, where sec f = {5 C X: S fl F / 0 (F € f)}. S' is finer than <5 if S' C S. The supremum supá,- (with respect to this relation “finer”) of the proximities Si does exist. t The supremum of Riesz/Lodato proximities has the same property. A cover 1980 Mathematics Subject Classification (1985 Revision). Primary 54E15; Secondary 54E05. Key words and phrases. (Riesz/Lodato) proximity, (Riesz/Lodato) contiguity, proxi­­mally fine contiguity, compressed/Cauchy filter. Research supported by the Hungarian National Foundation for Scientific Research Grant No. 1807. 0081-6906/94/$ 4.00 ©1994 Akadémiai Kiadó, Budapest MAGYAR TUDOMÁNYOS AKADÉMIA KÖNYVTÁRA

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