Studia Scientiarium Mathematicarum Hungarica 29. (1994)

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

J. DEÁK c of X is a 6-cover if for any sets A and B with. A 6 B, there is a C G c that meets both A and B. 0.2 A contiguity on X is a non-empty collection T of finite covers of X such that (i) if cG T and c refines 5 then ö G T; (ii) if c,ö G T then there is an element of T that refines both c and D. (c refines t> if each element of c is contained in some element of ö.) 93 C T is a base for T if each element of T is refined by some element of 03; it is a subbase for T if {(0)3: 0 ^ F C 93, 5 is finite} is a base for T; here (0)3 is defined as follows: A G (0)3 iff there are A(c) G c such that A = 0{^(c): c e 3}; we shall write c(n)D for (f|){c>0}- is finer than T if T' D T. The contiguity T induces the proximity 6 = <5(T) for which AS B iff, for each c G T, there is a C G c with AilC/0^j5nC; in other words, T is compatible with 6. The contiguity T is Riesz if, for each c G T, int c = {int C: C G c} is a cover, where int C = X \ c(X \ C), with c defined for Ó(T); T is Lodato if int c G T whenever cG T. A Riesz/Lodato contiguity induces a Riesz/Lodato proximity. The supremum sup T, (with respect of the relation “finer”) of the contiguities T, does exist, and, assuming i G I 0, U T,- is a subbase for it. S The supremum of Riesz/Lodato contiguities has the same property. A filter f on X is T-Cauchy if fflc ^ 0 (c G T). Any T-Cauchy filter is <$(r)-compressed. 0.3 Any proximity 6 can be induced by contiguities; T°(<5) is the coarsest and T1(á) the finest one, where the <5-covers of cardinality ^ 2, i.e. the covers c a,b = {X\A,X\B] (A6B), form a subbase for r°(6), while rx(i) consists of all the finite Á-covers. If S is Riesz or Lodato then so is T°(£); the finest compatible Riesz contiguity r}*(<5), respectively Lodato contiguity 1^(0), can be described as follows: c G G T}j(ő) iff cis afinite £-cover and int c is a cover; c G T^(i) iff c is a finite cover and int c is a Á-cover. T°, T1, and T^ are functors from the category of (Riesz/Lodato) proximities into the category of (Riesz/Lodato) contiguities; this simple fact is, however, irrelevant from the point of view of the present paper. A cover c of X will be called strong if the collection {X \ C: C G c} is disjoint. If |c| <2 then c is evidently strong. If c is strong and |c| > 3 then c is a <$-cover for any proximity 6. § 1. Suprema of proximally fine contiguities If S' is finer than 6 then T1(i/) is evidently finer than rx(Á). Similar statements hold for T^ and T^: check that, with int' understood in the

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