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Nonstandard Topologies with Bases That Consist Only of Standard Sets

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Nonstandard Topologies with Bases That Consist Only of Standard Sets PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 129, Number 7, Pages 2075{2083 S 0002-9939(00)05790-7 Article electronically published on December 4, 2000 NONSTANDARD TOPOLOGIES WITH BASES THAT CONSIST ONLY OF STANDARD SETS NADER VAKIL (Communicated by Carl G. Jockusch, Jr.) Abstract. Let X be an infinite set, D a set of pseudo-metrics on X; Ξ ⊂ ∗X; and Γ ⊂ ∗D: If ρ(a; b) is limited (finite) for every a; b 2 Ξ and every ρ 2 Γ; then, for each ρ 2 Γ; we can define a pseudo-metricρ ~ on Ξ by writingρ ~(a; b)=st(ρ(a; b)): We investigate the conditions under which the topology induced on Ξ by fρ~ : ρ 2 Γg has a basis consisting only of standard sets. This investigation produces a theory with a variety of applications in functional analysis. For example, a specialization of some of our general results will yield such classical compactness theorems as Schauder's theorem, Mazur's theorem, and Gelfand-Philips's theorem. 1. Introduction Let X; D; Ξ and Γ be as in the abstract. Suppose that ◦Ξ=fx 2 X : ∗x 2 Ξg6= ; and ◦Γ=fd 2 D : ∗d 2 Γg6= ;.Letρ(a; b) be limited (finite) for every a; b 2 Ξandρ 2 Γ, and let Γ=~ fρ~ : ρ 2 Γg whereρ ~(a; b)=st(ρ(a; b)): Let (Ξ; Γ)~ denote the uniform space whose uniformity is generated by the set Γ.~ The topology induced on Ξ by this uniformity will be referred to as Γ-topology~ on Ξ. We call Γ compatible with Ξ if the Γ-topology~ on Ξ has a basis that consists only of standard sets. In this paper we investigate the following question. Question. Under what conditions on the sets Ξ and Γ will the set Γ be compatible with Ξ? Our main results regarding this question are presented in Section (2), where we treat the question in a setting that is suitable for a wide range of applications. In Section (3), after giving several nonstandard compactness criteria for subsets of a Banach space and its dual, we use the theory developed in Section (2) to derive such classical compactness theorems as Schauder's theorem, Mazur's theorem, and Gelfand-Philips's theorem. The idea of the Γ-topology~ introduced above is an extension of the notion of S-topology introduced by Robinson [1] in the context of metric spaces, and further studied by Henson ([2], page 117) in the context of uniform spaces. What was considered in [2] was a special case of (Ξ; Γ)~ in which Ξ = ∗X and Γ=f ∗d : d 2 Dg. In this special case, we shall call the Γ-topology~ the S- topology although it was not called so in [2]. The question of existence of a basis Received by the editors June 8, 1999 and, in revised form, November 16, 1999. 2000 Mathematics Subject Classification. Primary 46S20; Secondary 54J05. Key words and phrases. Nonstandard topologies, Banach space, compactness conditions. c 2000 American Mathematical Society 2075 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 2076 NADER VAKIL consisting only of standard sets for the S-topology was not considered by Robinson or by Henson. Topologies that are generated by a family of standard sets on ∗X should not be confused with the S-topology as defined in [1] and [2] and generalized to Γ-topology~ here. A special case of the former was first introduced in [4], page 47, and called the discrete S-topology. The idea was somewhat extended and further discussed in [3], page 197, where the relation of such topologies to compactifications of X was pointed out. In [6], starting with a topological space (X; T), Salbany and Todorov considered the topology T~ that is generated on ∗X by the collection f ∗G : G 2 T g, and used this topology to study various classical compactifications of (X; T). In [7], H. Render, starting with a uniform space (X; V) and its induced topo- logical space (X; T), studied the relation between the T~ (as defined in the above paragraph) and the S-topology (as defined on the previous page). The result he announced was that: if (X; V) is not zero-dimensional, then there does not exist a base of the S-topology consisting only of standard sets. But this statement is false. As a trivial example, consider the set X =(0; 1) with its usual unifor- + mity (i.e., the one which has the sets U = fhx; yi : jx − yj <g, 2 R ,fora basis). By definition, the S-topology associated with this uniformity has the sets S(a; )=fb 2 ∗X :stja − bj <g, a 2 ∗X; 2 R+, for a basis. It is obvious that the standard sets ∗S(x; )withx 2 X and 2 R+ also form a basis for the S-topology. This does not, however, happen in an arbitrary uniform space. The reason it happened here was that the set X =(0; 1) with its usual uniformity is a totally bounded space. In Corollary (1.3), we show that, given a uniform space (X; V), the S-topology on ∗X has a basis consisting only of standard sets if and only if (X; V) is totally bounded. Thus we first address our opening question in the general setting where it is posed. To fix the notation, let (X; D) be a uniform space with its uniformity generated by a set of pseudo-metrics D. If, for an element a 2 ∗X, ∗d(a; x)is limited for all d 2 D and all x 2 X,thenwesaythata is limited. We denote the set of all limited elements of ∗X by ltd( ∗X). For other nonstandard concepts used in this paper, we refer the reader to [3]. The reference [5] contains all of the standard concepts used in this paper. Theorem 1.1. Let Γ=f ∗d : d 2 Dg. A subset A of X is totally bounded in (X; D) if and only if ∗A ⊂ ltd( ∗X) and the Γ~-topology on ∗A has a basis consisting only of standard sets. Proof. Recall that A is totally bounded if and only if every element of ∗A is pre-near standard. Assume that A is totally bounded. Since ∗A ⊂ pns( ∗X) ⊂ ltd ( ∗X), ∗d(a; b) is limited for every d 2 D and every a; b 2 ∗A.Furthermore, for all a 2 ∗A; x 2 X; 2 R+; and d 2 D,thesets (1) S~(a; x; d; )=fb 2 ∗A :st(j ∗d(x; b) − ∗d(x; a)j) <g are open relative to the Γ-topology,~ are standard, and form a subbasis for the Γ-topology~ on ∗A. To prove they are open, fix b 2 S~(a; x; d; ). Choose δ = − st(j ∗d(x; b) − ∗d(x; a)j). Since S~(b; d; δ) ⊂ S~(b; x; d; δ), we have S~(b; d; δ) ⊂ S~(a; x; d; ). To see that they are standard, note that the expression st(j ∗d(x; b) − ∗d(x; a)j) in(1)canbewrittenas ja^(x; d) − ^b(x; d)j,where c^ : X × D ! R (c 2 ltd( ∗X)) is defined byc ^(x; d)=st(∗d(c; x)). Thusthe License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use NONSTANDARD TOPOLOGIES 2077 defining formula of S~(a; x; d; ) is equivalent to one that contains only standard constants. This proves that the set S~(a; x; d; ) is standard. Next let S~(a; d; )= fb 2 ∗A :st(∗d(a; b))<g be a subbasic element of the Γ-topology~ of ∗ 2 ∗ A.Choosex X such that st( d(a; x))< 3 . Then it is easily seen ~ ⊂ ~ that S(a; x; d; 3 ) S(a; d; ). This proves that the sets defined in (1) form a subbasis for the Γ-topology~ on ∗A. The converse is an easy consequence of the definitions. In the next two corollaries we continue to assume that Γ = f ∗d : d 2 Dg. Since, for subsets of a complete uniform space, total boundedness and relative compactness are equivalent properties, Theorem (1.1) has the following immediate corollary. Corollary 1.2. Suppose that (X; D) is a complete uniform space. A subset A of X is relatively compact in (X; D) if and only if ∗A ⊂ ltd( ∗X) and the Γ~-topology on ∗A has a basis consisting only of standard sets. Corollary 1.3. Suppose that D consists only of bounded pseudo-metrics on X: The following assertions are equivalent. (i) The space (X; D) is totally bounded. (ii) The space (∗X; Γ)~ has a basis consisting only of standard sets. (iii) The space (∗X; Γ)~ is quasi-compact and has X as a dense subset. Proof. The implication (i) =) (ii) is an immediate consequence of Theorem (1.1). Assume that (ii) holds, and let B~ be a basis of (∗X; Γ)~ that consists only of standard sets. Since every nonempty member of B~ contains a standard element, X is dense in (∗X; Γ).~ The quasi-compactness of (∗X; Γ)~ follows from the fact C~ f ∗ n 2 Bg~ if is a nonemptyT subfamily of C : X C with the finite intersection property, then C6~ = ;: For the implication (iii) =) (i), we need only observe that the denseness of X in (∗X; Γ)~ implies that every point in ∗(X; D)is pre-nearstandard. 2. Main theorem Corollary (1.2) suggests that the notion of compatibility defined in the opening paragraph of the previous section might give rise to interesting compactness con- ditions in functional analysis.
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