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Strict Regular Completions of Cauchy Spaces 1549 PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 128, Number 5, Pages 1547{1554 S 0002-9939(99)05132-1 Article electronically published on October 5, 1999 STRICT REGULAR COMPLETIONS OF CAUCHY SPACES PAUL BROCK AND GARY RICHARDSON (Communicated by Alan Dow) Abstract. A diagonal condition is defined and used in characterizing the Cauchy spaces which have a strict, regular completion. 1. Introduction Cauchy spaces were introduced in order to study completions from a general point of view. The first study of completions resembling Cauchy spaces defined here seems to be due to Kowalsky [13]. Keller [9] gave the present definition of Cauchy spaces by characterizing the permissible set of Cauchy filters for a uniform convergence space introduced by Cook and Fischer [4]. Early efforts to develop a completion theory for uniform convergence spaces led to the realization that Cauchy spaces provide a more natural foundation, and subsequent research shifted to Cauchy spaces. Cauchy spaces have been used in diverse areas; for example, Ball [1], [2] made extensive use of Cauchy spaces in his work on completions of lattices and lattice ordered groups, and McKennon [17] applied the notion of a Cauchy space in his study of C∗-algebras. Applications of Cauchy spaces to the study of function spaces can be found in the monograph by Lowen-Colebunders [16]. Kowalsky [13] and Cook and Fischer [5] introduced diagonal axioms for conver- gence spaces. A convergence structure obeys Cook and Fischer's diagonal axiom if and only if it is topological, whereas, a pretopological convergence structure sat- isfying Kowalsky's diagonal condition must be topological. Moreover, Cook and Fischer [5] showed that a convergence space is regular if and only if it obeys the \dual" of their diagonal axiom. Kent and Richardson [12] introduced a diagonal condition for Cauchy spaces and proved that a Cauchy space has a topological com- pletion exactly when it satisfies this diagonal axiom. Unfortunately, the \dual" of the above mentioned diagonal condition fails to imply that the Cauchy space com- pletion, or even the underlying Cauchy space, is regular. The dual of the diagonal axiom introduced here is used in characterizing when a strict, regular completion exists. The study of regular completions of Cauchy spaces was initiated by Rama- ley and Wyler [19]. Regularity of Kowalsky's completion was characterized by E. Lowen [14]. Received by the editors September 23, 1997 and, in revised form, July 7, 1998. 1991 Mathematics Subject Classification. Primary 54A20, 54D35. Key words and phrases. Convergence space, Cauchy space, regularity, completion, diagonal axiom. c 2000 American Mathematical Society 1547 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 1548 PAUL BROCK AND GARY RICHARDSON 2. Preliminaries Let X be a set, F(X) the set of all (proper) filters on X,and2X the set of all subsets of X.Forx 2 X,denoteby_x the fixed ultrafilter generated by fxg.A filter F is finer (coarser ) than a filter G if G⊆F(F⊆G) and is denoted by F≥G(G≤F). Given F, G2F(X), the supremum of F and G, denoted F_G, exists provided F \ G =6 ; for each F 2F, G 2G, and is defined to be the smallest filter containing both F and G. Definition 2.1. A convergence structure q on a set X is a function q : F(X) ! 2X satisfying: (C1) x 2 q(_x), for all x 2 X; (C2) F≤Gimplies q(F) ⊆ q(G); (C3) x 2 q(F) implies x 2 q(F\x_). The pair (X; q) is called a convergence space,andx 2 q(F) is interpreted to q mean \F q-converges to x," and is usually written \F ! x." If q is a convergence structure on X, then a function f :(X; q) ! (Y;p) between convergence spaces is p q continuous provided f(F) ! f(x) whenever F ! x.Ifp and q are convergence structures on X and f :(X; q) ! (X; p) is continuous, where f is the identity map on X,thenwewritep ≤ q (p is coarser than q,orq is finer than p). The closure operator clq and interior operator Iq associated with each conver- gence space (X; q) are defined for each A 2 2X as follows: q clqA = fx 2 X : F ! x for some A 2Fg; q IqA = fx 2 A : F ! x implies A 2Fg. If F is a filter on X, clqF denotes the filter whose base is fclqF : F 2Fg,and q q a convergence space (X; q) is called regular if F ! x implies clqF ! x. For every x 2 X,theq-neighborhood filter at x is defined to be Vq(x)=fV ⊆ X : x 2 IqV g; and A ⊆ X is called q-open when IqA = A.Moreover,q is called a pretopology if q Vq(x) ! x for each x 2 X, and a pretopology is a topology if each Vq(x) has a base of q-open sets. Cook and Fischer [5] defined the following two diagonal conditions for conver- gence spaces in terms of a \compression operator" for filters. These conditions, denoted by F and F 0, are dual to each other in the sense that F 0 may be obtained from F by reversing the implication in the last sentence of F , and vice-versa. Let X and Y be non-empty sets, F2F(Y ), and σ : Y ! F(X). Define [ \ KσF = σ(y); F 2F y2F K is called the \compression operator for F relative to σ." In order to avoid repe- tition, let us name the following statement. ∆ : Suppose that (Z; r) is a convergence space containing (X; q) as a subspace. g r Assume that : Y ! Z and σ : Y ! F(X)satisfiesσ(y) ! (y)foreachy 2 Y , g where σ(y) denotes the filter on Z whose basis is σ(y). Then the couple (ψ,σ)is said to be an exterior selection of (Z; r). Given a convergence space (X; q), diagonal conditions F and F 0 can be stated in terms of ∆ with (Z; r)=(X; q). License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use STRICT REGULAR COMPLETIONS OF CAUCHY SPACES 1549 q F : For every exterior selection (ψ,σ)of(X; q)andeachF2F(Y ), F ! x q implies that KσF ! x. q F 0 : For every exterior selection (ψ,σ)of(X; q)andeachF2F(Y ), KσF ! x q implies that F ! x. Important results pertaining to these two axioms are summarized below. Theorem 2.2. Let (X; q) be a convergence space. (1) [7] (X; q) is topological if and only if it satisfies F . (2) [5] (X; q) is regular if and only if it satisfies F 0. Definition 2.3. A Cauchy structure C on a set X is a collection of filters on X satisfying: (CHY1)_x 2C, for all x 2 X; (CHY2) F2Cand F≤Gimplies G2C; (CHY3)IfF; G2Cand F_Gexists, then F\G2C. Apair(X; C) consisting of a set X and a Cauchy structure C on X is called a Cauchy space. For each Cauchy space (X; C), there is an associated convergence F !qC F\ 2C C structure qC on X defined by x iff x_ .ACauchyspace(X; )isT2 (or Hausdorff )if(X; qC )isT2 in the sense that each qC -convergent filter has a unique C F2C F2C limit. A Cauchy space (X; )isregular if clqC whenever . It should C be mentioned that (X; qC ) is regular whenever (X; ) is regular, but not conversely. C F C ACauchyspace(X; ) is called complete if every filter in is qC -convergent. Keller [9] showed that a convergence space (X; q) is induced by some (complete) Cauchy space iff for each distinct pair of points x, y 2 X, the set of all filters which q-converge to x is either equal to or disjoint from the set of all filters which q-converge to y. Indeed, a sufficient condition occurs when (X; q) is Hausdorff. Let (X; C) be a Cauchy space. An equivalence relation ∼ on C is defined as follows: For F; G2C, F∼Giff F\G2C.IfF2C,let[F]C = fG 2 C : F∼Gg be the equivalence class determined by F; this equivalence class is denoted simply by [F] if there is no ambiguity. Given Cauchy spaces (X; C)and(Y;D), a map f :(X; C) ! (Y;D)isCauchy continuous if F2Cimplies f(F) 2D.If,in addition, f :(X; C) ! (Y;D) is a bijection such that f and f −1 are both Cauchy continuous, then f is a Cauchy isomorphism.Let(X; C) be a Cauchy space and A ⊆ X.AfilterF2F(X)hasatrace on A if F \ A =6 ;,foreachF 2F.Inthis case, FA = fF \ A : F 2Fgdenotes the trace of F on A; CA = fFA : F2C; F has a trace on Ag is a Cauchy structure on A,and(A; CA)isaCauchy subspace of (X; C). Given a Cauchy space (X; C), the triple (Y;D,θ) is called a completion of (X; C) if (1) (Y;D) is complete; C ! D (2) θ :(X; ) (θ(X); θ(X) ) is a Cauchy isomorphism; (3) clqD θ(X)=Y: Moreover, the completion (Y;D,θ)issaidtobestrict if for each H2Dthere F2C F ≤H exists such that clqD θ( ) : Several closely related diagonal axioms for a Cauchy space are considered. Let ∗ (X; C) be a Cauchy space, N the set of all F2Cwhich fail to qC-converge, X = X [f[F]:F2Ng, and define j : X ! X∗ by j(x)=x for each x 2 X.
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