ℒ-Realcompactifications and Normal Bases

Alò, Richard A.; Shapiro, Harvey L.

doi:10.1017/s1446788700007448

Cited by 19 publications

(5 citation statements)

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“…Now let v e IR(σ, ^f 2 ). v can be extended to a p 6 IR(σ, £f 4 Proof. This follows from the previous corollary observing that countably paracompact and normal is preserved under continuous closed surjections.…”

Section: ]) If X Is a Realcompact Tyckonoff Space And Eczx Is In τσ(mentioning

confidence: 99%

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Regular lattice measures: mappings and spaces

Bachman¹,

Sultan²

1976

Pacific J. Math.

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We prove in this paper a very general measure extension theorem which has as corollaries many recent, significant extension theorems in the literature. We apply these results to the question of when there is a well behaved map from the σ-smooth lattice regular measures on one set to the σ-smooth lattice regular measures on a second set. After developing these general theorems we specialize consideration to two valued latitce regular measures and obtain in a new and consistent manner many important mapping and subspace theorems on the preservation of different types of repleteness including results of Dykes, Hager, Isiwata, Moran, Varadarajan, Gillman, Jerrison and others. Frolik [19], and others, we have expressed measure theoretically in terms of two valued .Sf-regular measures. This is advantageous; for besides being analytically simpler to work with than with filters, many theorems in this form can be generalized naturally to arbitrary irregular measures, and the topological settings extended from Wallman topologies to vague topologies.The notion of ^-replete includes as special cases: real compact, Borel complete [24], α-complete [15], etc., and in [6], we developed measure-theoretic results to show systematically how to obtain repleteness interrelations. Here, we are concerned with mapping and subspace problems. The mapping questions are of the type: Given T: X-• Y which is well-behaved (see §5 for details) with respect to two lattices £έ\ y J^ of subsets of X and Y respectively, when does T induce a well-behaved mapping T**: MR{σ, £? ^-*MR{σ, & §> where, in general, MR(σ, £f) designates the σ-smooth i5f-regular measures on a set X with respect to a lattice, ^ of subsets? We develop the major results of this type in §5, and when applied to the special case of two-valued measures, we get as corollaries important subspace and mapping results of Frolik [21]

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Section: ]) If X Is a Realcompact Tyckonoff Space And Eczx Is In τσ(mentioning

confidence: 99%

“…If ^ is a disjunctive separating lattice this is equivalent to demanding that each μeIR (.2f) is concentrated at a point; i.e., that IR(σ, £f) is the set of all degenerate measures. Frolik [19] uses maximally complete, while some authors just use the word complete (see [4], [24]). In the case <£?…”

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confidence: 99%

Regular lattice measures: mappings and spaces

Bachman¹,

Sultan²

1976

Pacific J. Math.

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“…However many important results and properties pertaining to the Stone-Cech compactification and the Hewitt realcompactification can be extended to a more general setting by considering appropriate lattices of sets, generalizing that of the lattice of zero sets in a Tychonoff space. This program was first considered by Wallman (1938) and Alexandroff (1940) and has more recently appeared in Alo and Shapiro (1970), Banachewski (1962), Brooks (1967), Frolik (1972), Sultan (to appear) and others.…”

Section: Introductionmentioning

confidence: 99%

General rings of functions

Sultan

1975

J. Aust. Math. Soc.

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Several authors have studied various types of rings of continuous functions on Tychonoff spaces and have used them to study various types of compactifications (See for example Hager (1969), Isbell (1958), Mrowka (1973), Steiner and Steiner (1970)). However many important results and properties pertaining the Stone-Čech compactification and the Hewitt realcompactification can be extended to a more general setting by considering appropriate lattices of sets, generalizing that of the lattice of zero sets in a Tychonoff space. This program was first considered by Wallman (1938) and Alexandroff (1940) and has more recently appeared in Alo and Shapiro (1970), Banachewski (1962), Brooks (1967), Frolik (1972), Sultan (to appear) and others.

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“…on A determines a Txextension of A which we now proceed to describe. These extensions have been studied previously by R. A. Alo and H. L. Shapiro [1], [2], A. K. Steiner and E. F. Steiner [8], and M. S. Gagrat and S. A. Naimpally [5] with the additional hypothesis that «á? is normal, in which case X is Tychonoff.…”

mentioning

confidence: 99%

“…It is easy to see that A is =SP-realcompact iff for every a e A*, there exists x e X such that {x} e a, and this condition is equivalent to A being homeomorphic to A* by the map ex: X->X*. R. A. Alo and H. L. Shapiro [1] proved (and their proof is valid in our more general setting) that X* is JSP*-realcompact, and they call A* the JSP-realcompactification of A (by a slight abuse of language). Our principal objective is to find à category for which r¡(X, JSP) = A* becomes an epireflection functor.…”

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confidence: 99%

ℒ-realcompactifications as epireflections

Bentley¹,

Naimpally²

1974

Proc. Amer. Math. Soc.

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If if is a countably productive normal base on a Tychonoff space X, then r¡(X,-a*) is an-Sf^-realcompact extension of X. R. A. Alo and H. L. Shapiro thus generalized the Hewitt realcompactification of X. In the following paper, we extend this construction to ^-spaces and show that it is an epireflection functor on an appropriate category. We are thus concerned with the question of the extendibility of a continuous map/: X-+Y\.o a continuous map g:r¡(X,¿£'x)->-rj(Y,J?'r). We derive necessary and sufficient conditions therefor in the case when Sfr is a nest generated intersection ring on Y. All of the topological spaces which we consider are assumed to satisfy the Fj-separation axiom. A family =5? of closed subsets of a space X is called a countably productive separating base if the following are satisfied: (i) J? is a ring, i.e. it is closed under finite unions and finite intersections and 0,Xe^. (ii) ¿£ is countably productive, i..e closed under countable intersections. (iii) =Sf is separating, i.e. whenever x e X-S, S closed in X, there exist Lx, L2eJ¡? such that x e Lx, 5c L2, and LxC\L2=0 (E. F. Steiner [9]). If in addition to the above properties, (iv) =S? is nest generated, i.e. L e =£? implies the existence of En, Fn e 3?, neN, such that Fn+1^X-Ena Fn and L=f) {Fn:n e N} (A. K. Steiner and E. F. Steiner [8], [10]), then J? is called a nest generated intersection ring. (R. A. Alo and H. L. Shapiro [2] use the term strong delta normal base.) In [8], A. K. Steiner and E. F. Steiner prove that a nest generated intersection ring =Sf is normal, i.e. Lx, L2 e^C, and LxC\L2=0 implies the existence of L'x, L!2e££ such that LxnL[=0, L2C\L'2=0, and L'XKJL'2 = X. A well-known example of a nest generated intersection ring is Z(X), the family of all zero sets of a Tychonoff space X.

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ℒ-Realcompactifications and Normal Bases

Cited by 19 publications

References 3 publications

Regular lattice measures: mappings and spaces

Regular lattice measures: mappings and spaces

General rings of functions

ℒ-realcompactifications as epireflections

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