Generalized reflective cum coreflective classes in Top and Unif

The Herrlich's problem from [8] whether there are nontrivial classes of topological spaces that are both almost reflective or injective and almost coreflective or projective, is investigated in a more general setting using cone and cocone modifications of the classes used in the problem. We look also at the problem for uniform spaces. Typical results: There is no nontrivial multiprojective and orthogonal class of topological spaces; There is a reflective class of uniform spaces that is almost coreflective in Unif.Work on this paper was initiated while the first author was a C.N.R. visitor of the University of L'Aquila. Partial financial assistence by Charles University Grant 349/1994 is also acknowledged.     

On the largest coreflective Cartesian closed subconstruct of Prtop

We show that the subconstruct Fing of Prtop, consisting of all finitely generated pretopological spaces, is the largest Cartesian closed coreflective subconstruct of Prtop. This implies that in any coreflective subconstruct of Prtop, exponential objects are finitely generated. Moreover, in any finitely productive, coreflective subconstruct, exponential objects are precisely those objects of the subconstruct that are finitely generated. We give a counterexample showing that without finite productivity the previous result does not hold.     

Cartesian-closed coreflective subcategories of uniform spaces generated by classes of metric spaces

The main result, in Theorem 3, is that in the category Unif of Hausdorff uniform spaces and uniformly continuous maps, the coreflective hulls of the following classes are cartesian-closed: all metric spaces having no infinite uniform partition, all connected metric spaces, all bounded metric spaces, and all injective metric spaces.Furthermore, Theorems 1 and 4 imply that if $\text{C}$ is any coreflective, cartesian-closed subcategory of Unif in which enough function space structures are finer than the uniformity of uniform convergence (as in the above examples), then either (1) $\text{C}$ is a subclass of the locally fine spaces, or (2) $\text{C}$ contains all injective metric spaces and $\text{C}$ is a subclass of the coreflective hull of all uniform spaces having no infinite uniform partition.     

Function spaces in metrically generated theories

The theory of metrically generated constructs provides us with an excellent setting for the study of function spaces. In this paper we develop a function space theory for metrically generated constructs and, by considering different metrically generated constructs, we capture interesting examples. For instance, for uniform spaces we retrieve the uniformity of uniform convergence and its generalization to Σ-convergence and for UG-spaces we obtain a quantified version of these structures. Our theory also allows for many applications, in particular we are able to characterize the complete subspaces of these function spaces and we succeed in producing an appropriate Ascoli theorem.     

Separation properties in algebraic categories of topological spaces

Full subcategories C ? Top of the category of topological spaces, which are algebraic over Set in the sense of Herrlich [2], have pleasant separation properties, mostly subject to additional closedness assumptions. For instance, every C-object is a T₁-space, if the two-element discrete space belongs to C. Moreover, if C is closed under the formation of finite powers in Top and even varietal [2], then every C-object is Hausdorff. Hence, the T₂-axiom turns out to be (nearly) superfluous in Herrlich's and Strecker's characterization of the category of compact Hausdorff spaces [1], although it is essential for the proof.If we think of C-objects X as universal algebras (with possibly infinite operations), then the subalgebras of X form the closed sets of a compact topology on X, provided that the ordinal spaces [0, β] belong to C. This generalizes a result in [3]. The subalgebra topology is used to prove criterions for the Hausdorffness of every space in C, if C is only algebraic.     

ON CERTAIN FACTORIZATIONS OF FUNCTORS INTO THE CATEGORY OF QUASI-UNIFORM SPACES

This paper is motivated by the search for natural extensions of classical uniform space results to quasi-uniform spaces. As instances of such extensions we restate some theorems of P. Fletcher and W.F. Lindgren [Pacific J. Math. 43 (1971), 619–6311 on transitive quasi-uniformities and of S. Salbany [Thesis, Univ. Cape Town, 1971] on compactification and completion. The theorems as restated describe properties of certain right inverses of the functor which forgets the quasi-uniform structure and retains one induced topology (for Fletcher and Lindgren's work), respectively retains both induced topologies (for Salbany's work). Accordingly we investigate systematically the process by which the right inverses of the forgetful functors can be extended from the classical setting to one of these settings, and from one of these to the other.     

Quotient-reflective and bireflective subcategories of the category of preordered sets

In previous papers, the notions of “closedness” and “strong closedness” in set-based topological categories were introduced. In this paper, we give the characterization of closed and strongly closed subobjects of an object in the category Prord of preordered sets and show that they form appropriate closure operators which enjoy the basic properties like idempotency (weak) hereditariness, and productivity.We investigate the relationships between these closure operators and the well-known ones, the up- and down-closures. As a consequence, we characterize each of T₀, T₁, and T₂ preordered sets and show that each of the full subcategories of each of T₀, T₁, T₂ preordered sets is quotient-reflective in Prord. Furthermore, we give the characterization of each of pre-Hausdorff preordered sets and zero-dimensional preordered sets, and show that there is an isomorphism of the full subcategory of zero-dimensional preordered sets and the full subcategory of pre-Hausdorff preordered sets. Finally, we show that both of these subcategories are bireflective in Prord.     

Asymmetry and bicompletion of approach spaces

We develop a bicompletion theory for the category Ap₀ of T₀ approach spaces in the sense of Lowen [R. Lowen, Approach Spaces: The Missing Link in the Topology-Uniformity-Metric Triad, Oxford University Press, Oxford, 1997], which extends the completion theory obtained in [R. Lowen, K. Robeys., Completions of products of metric spaces, Quart. J. Math. Oxford 43 (1991) 319-338] for the subcategory of Hausdorff uniform approach spaces. Moreover, we prove it to be firmly epireflective (in the sense of [G.C.L. Brümmer, E. Giuli, A categorical concept of completion of objects, Comment. Math. Univ. Carolin. 33 (1992) 131-147]) with respect to a certain morphism class of dense embeddings.     

On categorical notions of compact objects

Due to the nature of compactness, there are several interesting ways of defining compact objects in a category. In this paper we introduce and study an internal notion of compact objects relative to a closure operator (following the Borel-Lebesgue definition of compact spaces) and a notion of compact objects with respect to a class of morphisms (following Áhn and Wiegandt [2]). Although these concepts seem very different in essence, we show that, in convenient settings, compactness with respect to a class of morphisms can be viewed as Borel-Lebesgue compactness for a suitable closure operator. Finally, we use the results obtained to study compact objects relative to a class of morphisms in some special settings.Partial financial assistance by Centro de Matemática da Universidade de Coimbra and by a NATO Collaborative Grant (CRG 940847) is gratefully acknowledged.     

On sequentially compact T2 compactifications

It is known that a compact space can fail to be sequentially compact. In this paper we consider the following problem: when does a space admit a sequentially compact T₂ compactification? In the first section we develop a method to produce such compactifications, and we apply it in the second section to study the question using coverings.Moreover, we obtain solutions for locally compact T₂ spaces, and for metrizable spaces.     

An elementary approach to ‘Algebra ∩ topology = compactness’

Herrlich and Strecker characterized the category Comp ₂ of compact Hausdorff spaces as the only nontrivial full epireflective subcategory in the category Top ₂ of all Hausdorff spaces that is concretely isomorphic to a variety in the sense of universal algebra including infinitary operations. The original proof of this result requires Noble's theorem, i.e. a space is compact Hausdorff iff every of its powers is normal, which is far from being elementary. Likewise, Petz' characterization of the class of compact Hausdorff spaces as the only nontrivial epireflective subcategory of Top ₂, which is closed under dense extensions (= epimorphisms in Top ₂) and strictly contained in Top ₂ is based on a result by Kattov stating that a space is compact Hausdorff iff its every closed subspace is H-closed. This note offers an elementary approach for both, instead.Presented at the European Colloquium of Category Theory, Tours, France, 25–31 July 1994.     

Lexicographic sums and fibre-faithful maps

The notion of lexicographic sum is introduced in general categories. Existence criteria are derived, particularly for locally cartesian closed categories and for categories with suitable coproducts. Lexicographic sums satisfy a generalized associative law. More importantly, every morphism can be factored through the lexicographic sum of its fibres. This factorization and the two types of maps arising from it, fibre-trivial and fibre-faithful, are studied particularly for partially ordered sets and forT ₁-spaces.     

LEGENDRE TRANSFORMATIONS AND CARTAN FORMS IN THE CALCULUS OF VARIATIONS OF MULTIPLE INTEGRALS

ABSTRACT

(PART II): In terms of a given Hamiltonian function the 1-form w = dH + ?_j|dπ^j is defined, where {?_j:j = 1,…, n} denotes an invariant basis of the planes of the distribution D_n. The latter is said to be canonical if w = 0 (which is analogous to the definition of Hamiltonian vector fields in symplectic geometry). This condition is equivalent to two sets of canonical equations that are expressed explicitly in term of the derivatives of H with respect to its positional arguments. The distribution D_n is said to be pseudo-Lagrangian if dπ^j(?_j,V_h) = 0; if D_n, is both canonical and pseudo-Lagrangian it is integrable and such that H = const. on each leaf of the resulting foliation. The Cartan form associated with this construction [9] is defined a II = π² ? ? πⁿ. If π is closed, the distribution D_N is integrable, and the exterior system {π^j} admits the representation ψ^j = dS^j in terms of a set of 0-forms S^j on M. If, in addition, the distribution D_N is canonical, these functions satisfy a single first order Hamilton-Jacobi equation, and conversely. Finally, a complete figure is constructed on the basis of the assumptions that (i) the Cartan form be closed, and (ii) that the distribution D_n, be both canonical and integrable. The last of these requirements implies the existence of N functions ψ^A that depend on x^h and N parameters w^B, whose derivatives are given by ?ψ^A (x^h, w^B)/?x^j = B^A _j (x^h, ψ^B (x^h,w^B)). The complete figure then consists of two complementary foliations: the leaves of the first are described by the functions ψ^A and satisfy the standard Euler-Lagrange equations, while the second, that is, the transversal foliation, is represented by the aforementioned solution of the Hamilton-Jacobi equation. The entire configuration then gives rise in a natural manner to a generalized Hilbert independent integral and consequently also to a generalized Weierstrass excess function.     

DISCONNECTEDNESS

Abstract

There are three different ways to characterize To-spaces in the category of topological spaces. All three methods are canonical, i.e. they can be easily formulated in a general setting, where they, in general, do not coincide. In the following, the characterization of T₀-spaces by indiscrete spaces is generalized to an abstract category and investigated.     

Extremal Prime Filters and Universality of Some Categories

Abstract

Due to the existence of constants, classical topological categories cannot be universal in the sense of containing each concrete category as a full subcategory. In the point-free case, this obstruction vanishes and the question of universality makes sense again. The main problem, namely that as to whether the category of locales and localic morphisms is universal is still open; we prove, however, the universality of the following categories:

- pairs (locale, sublocale) with the localic morphisms preserving the distinguished sublocales,

- frames with frame homomorphisms reflecting the maximal prime ideals,

- Priestley spaces with f-maps preserving the maximal elements.     

SOME MODULES OVER A B*-ALGEBRA

Abstract

Some modules, ?_p (B) (1 < p < ∞), over a commutative B*-alge= bra with identity are discussed and these are seen to have properties similar to the Banach spaces ?_p (1 < p < ∞).     

Are there convenient subcategories of Top?

The category Top has several pleasant properties but fails to have other desirable ones. Consequently there have been various attempts to replace Top by more convenient categories; mostly subcategories or supercategories of Top. Whereas several of the supercategories of Top have extremely pleasant properties, all the subcategories of Top investigated so far have some deficiencies. In the present article it is shown why this is so and why the search for more convenient subcategories of Top has to be in vain. As a preparation in Section 1 nine convenience-properties for topological categories are defined and their mutual relations are analyzed. In Section 2 it is shown that every topological subcategory of Top (under some minor, natural assumptions) it-with respect to each of the 9 mentioned properties-as convenient or inconvenient as Top itself.     

Interior operators and topological separation

A notion of separation with respect to an interior operator in topology is introduced and some basic properties are presented. In particular, it is shown that this notion of separation with respect to an interior operator gives rise to a Galois connection between the collection of all subclasses of the class of topological spaces and the collection of all interior operators in topology. Characterizations of the fixed points of this Galois connection are given and examples are provided.