Anthony W. Hager

Minimal Covers of Topological Spaces

“Space” means topological space, always Hausdorff and frequently Tychonoff. A “cover” of the space X is a preimage of X by a perfect irreducible (PI) map. It is known that, for certain (rather similar) topological properties P, each space has a cover minimum for having P: we say, then, that P is a “covering property”. The first example of this was P = extremal disconnectivity (ED) and, after that, several others arose (in fact, quite naturally). This paper is an attempt a t a systematic theory of minimum covers and covering properties, emphasizing generality and simplicity.

On the localic Yosida representation of an archimedean lattice ordered group with weak order unit

W is the category of archimedean lattice ordered groups with weak order unit, and F is the category of frames. C:F→W is the functor which assigns to a given frame F the W-object CF=HomF(OR,F), where OR designates the frame of open sets of the real numbers R, and assigns to the frame homomorphism f:F→L the W-homomorphism Cf:CF→CL defined by Cf(g)=fg for all g ϵ CF. On the other hand, Y:W → F is the functor which assigns to a given W-object G the frame YG of W-kernels of G, and assigns to the W-homomorphism u:G→H the frame homomorphism Yu:YG→YH whose value at K ϵ YG is the W-kernel of H generated by u(K). We prove in Theorem 2.3.2 that Y is left adjoint to C, and in Theorem 2.4.3 that the adjunction restricts to an equivalence between the full subcategory RL of regular Lindelof frames and the full subcategory C of W-objects of the form CF for some frame F. The equivalence was first established by Madden and Vermeer. The unit μ=(μG) is the reflection of W in C, herein termed the (localic) Yosida representation, while the counit λ=(λF) is the (frame counterpart of the) reflection of locales into regular Lindelof locales. Finally, we note that the sense in which the Yosida representation is unique is precisely that (μG, YG) is the C-universal map for G.

Patch-generated Frames and Projectable Hulls

This article considers coherent frame homomorphisms h : L → M between coherent frames, which induce an isomorphism between the boolen frames of polars, with M projectable, and such that M is generated by h(L) and certain complemented elements of M. This abstracts the passage from a semiprime commutative ring with identity to its projectable hull. The frame theoretic setting is investigated thoroughly, first without any assumptions beyond the Zermelo–Fraenkel axioms of set theory, and, subsequently, assuming that algebraic frames are spatial. The culmination of this effort is the result that the spectrum of d-elements of M is obtained from that of L by refining the given hull–kernel topology to the patch topology. The second part of the article relates the projectable hull to the (von Neumann) regular hull, in a variety of contexts, including that of f-rings. For a uniformly complete f-algebra A, it is shown that the maximal ℓ-ideals of A that are traces of real maximal ideals of the regular hull HA are precisely the almost P-points of the space of maximal ℓ-ideals of A.

Hulls for Various Kinds of α-Completeness in Archimedean Lattice-Ordered Groups

Within Archimedean ℓ-groups, and with α an infinite cardinal or ∞, we consider X-hulls where X stands for any of the following classes of ℓ-groups: α-projectable; laterally α-complete; boundedly laterally α-complete; conditionally α-complete; combinations of the preceding, together with divisibility and/or relative uniform completeness. All these hulls exist, and may be obtained by iterated adjunction of the required extra elements, within the essential hull. When the ℓ-groups is relatively α-complemented one step in the iteration suffices for several crucial properties. We derive from the above a considerable number of equations involving combinations of these hull operators.

Notes on the Hewitt realcompactification of a product

Abstract Several theorems are presented on the relation υ(X × Y) = υX × υ Y; the proofs use the idea of z-embedding of a subspace. The main results are to the effect that this relation holds if it holds uniformly locally, e.g., if X × Y has a nonmeasurable normal cover by pseudocompact rectangles. These results use the following theorem: if X has a nonmeasurable normal cover U with υU ⊂ υX for each U ϵ U , then υX = ∪ {υU: Uϵ U } ; this immediately implies the Katětov–Shirota Theorem.

More on the Laterally σ-Complete Reflection of an Archimedean Lattice-Ordered Group

The laterally σ-complete reflection is given a new, one-step construction, which, in some notable cases affords an explicit realization. For example, if X is a compact Hausdorff topological space, the laterally σ-complete reflection of C(X) consists of all real-valued functions on X which are countably Baire-piecewise continuous. This realization of the reflection also shows, easily, that it carries rings to rings and, for objects with a designated unit, that the singularity of the unit is preserved.

Essential Completeness in Categories of Completely Regular Frames

The basic result here is that certain easily described coreflective subcategories S of the category CRFrm of completely regular frames have unique essential completions determined for each L ∈ S by the S-coreflection of the Booleanization of L. Next, this is shown to apply to several familiar subcategories of CRFrm, and concrete descriptions of the essential completions as well as internal characterizations of essential completeness are then given for these cases. Finally, back to the subcategories S in general, the essential completions in any of these are proved to become the epicomplete reflections in the category derived from S by considering only skeletal maps.

An α-disconnected space has no proper monic preimage

Abstract All spaces are compact Hausdorff. α is an uncountable cardinal or the symbol ∞. A continuous map τ:X→Y is called an α-SpFi morphism if τ-1(G) is dense in X whenever G is a dense α-cozero set of Y. We thus have a category α-SpFi (spaces with the α-filter) which, like any category, has its monomorphisms; these need not be one-to-one. For general α, we cannot say what the α-SpFi monics are, but we show, and R.G. Woods showed, that ∞-SpFi monic means range-irreducible. The main theorem here is: X has no proper α-SpFi monic preimage if and only if X is α-disconnected. This generalizes (by putting in α = ∞) the well-known fact: X has no proper irreducible preimage if and only if X is extremally disconnected. If, in our theorem, we restrict to Boolean spaces and apply Stone duality, we have the theorem of R. Lagrange, that in Boolean α-algebras, epimorphisms are surjective. The theory of spaces with filters has a lot of connections with ordered algebra—Boolean algebras of course, but also lattice-ordered groups and frames. This paper is a contribution to the development of this topological theory.

Characterization of Epimorphisms in Archimedean Lattice-Ordered Groups and Vector Lattices

This chapter is the first paper in a sequence of undetermined length, and this introduction is also a brief introduction to the sequence. An initial segment of that sequence is this chapter, followed by the essentially completed items listed in the bibliography as [Ball and Hager, a,b,c,d]. These papers treat various aspects of the theory of epimorphisms in the category Arch of archimedean l-groups with l-homomorphisms, and the category Wu of archimedean l-groups with distinguished weak unit and unit-preserving l-homomorphisms (and in the corresponding categories of vector lattices, which for all papers in the sequence can be disposed of with the comment: all this is true in vector lattices and the proofs require no change). Wu is itself interesting as a natural generalization of rings of continuous functions and as a setting for some functional analysis, and in any event seems to be a necessary bridge to Arch.

Specker spaces and their absolutes, I

Abstract A (Tychonoff) space is Specker if for each nonzero real-valued continuous function there is a nontrivial clopen set on which it is constant and nonzero. The following question is posed: if X is a Specker space, then is the absolute of X a Specker space as well? The answer is no, in general, but there are several interesting classes of spaces for which it is true. The dual notion of a Specker boolean algebra turns out to be equivalent to so-called (ω, 2)-distributivity, a fact which appears crucial in finding an example showing that the answer to the question raised above is indeed no. In the final section the totally ordered Specker spaces are characterized.