Exponential objects and Cartesian closedness in the constructPrtop

We give an internal characterization of the exponential objects in the constructPrtop and investigate Cartesian closedness for coreflective or topological full subconstructs ofPrtop. If $ is the set {0} {1/n;n 1} endowed with the topology induced by the real line, we show that there is no full coreflective subconstruct ofPrtop containing $ and which is Cartesian closed. With regard to topological full subconstructs ofPrtop we give an example of a Cartesian closed one that is large enough to contain all topological Fréchet spaces and allT ₁ pretopological Fréchet spaces.Aspirant NFWO     

α-Sober spaces via the orthogonal closure operator

Each ordinal equipped with the upper topology is a T ₀-space. It is well known that for =2 the reflective hull of in Top₀ is the subcategory of sober spaces. Here, we define -sober space for each 2 in such a way that the reflective hull of in Top₀ is the subcategory of -sober spaces. Moreover, we obtain an order-preserving bijective correspondence between a proper class of ordinals and the corresponding (epi)reflective hulls. Our main tool is the concept of orthogonal closure operator, first introduced in [12].The author acknowledges financial support from Instituto Politécnico de Viseu and from Centro de Matemática da Universidade de Coimbra.     

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.     

CompactT 0-spaces andT 0-compactifications

This paper deals with compactifications, particularly withk-completions, ofT ₀-spaces.The concept ofk-complete spaces, wherek is a cardinal with 1k , provides a useful gradation of compactness. The compact spaces are precisely the -complete spaces. For smallerk one obtains more restrictive concepts. However, for Hausdorff spaces, these concepts coincide for allk2. For eachk, a space< /i > < sub > < i > k < /i > < /sub > is constructed such that the < i > k < /i > -complete < i > T < /i > < sub > 0 < /sub > -spaces are precisely the extension closed subspaces of powers of < i > _k is constructed such that thek-completeT ₀-spaces are precisely the extension closed subspaces of powers of _k . Fork2 the associated$ _k -compactificationsX X areC*-embeddings. More generally, for a finite spaceE, allE-compactificationsX _E X areC*-embeddings if and only if the Sikorski spaces$ ₂ is a quotient of some connected closed subspace ofE.     

More on upper bicompletion-true functorial quasi-uniformities

Let T:QU₀→Top₀ denote the usual forgetful functor from the category of quasi-uniform T₀-spaces to that of the topological T₀-spaces. We regard the bicompletion reflector as a (pointed) endofunctor K:QU₀→QU₀. For any section F:Top₀→QU₀ of T we consider the (pointed) endofunctor R=TKF:Top₀→Top₀. The T-section F is called upper bicompletion-true (briefly, upper K-true) if the quasi-uniform space KFX is finer than FRX for every X in Top₀. An important known characterisation is that F is upper K-true iff the canonical embedding X→RX is an epimorphism in Top₀ for every X in Top₀. We show that this result admits a simple, purely categorical formulation and proof, independent of the setting of quasi-uniform and topological spaces. We thus mention a few other settings where the result is applicable. Returning then to the setting T:QU₀→Top₀, we prove: Any T-section F is upper K-true iff for all X the bitopology of KFX equals that of FRX; and iff the join topology of KFX equals the strong topology (also called the b- or Skula topology) of RX.     

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.     

Constant morphisms and constant subcategories

In a category supplied with a factorization system for morphisms and a fixed subcategory of constant objects, we introduce suitable notions ofconstant morphism and of the correspondingright andleft constant subcategories. The nature of constant morphisms we use captures two important features of constant subcategories: left-constant subcategories are right-constant in the dual category and the subcategory of constant objects contains relevant information on these subcategories. Furthermore, we present characterizations of constant subcategories in several contexts. Namely, we extend the characterization of disconnectednesses obtained by Huek and Pumplün, via terminal fans, to our context.The author acknowledges financial support by Centro de Matemática da Universidade de Coimbra.     

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.     

Some aspects of topological descent

The paper deals with (effective) descent morphisms for subfibrations X of the basic fibration Top/X, for topological spaces X and classes of continuous functions stable under pullback. For a category with pullbacks, we prove the stability under pullback of effective -descent morphisms for a class satisfying some suitable conditions. This plays a rôle in relating effective -descent to effective global-descent and enables us to obtain a criterion for effective étale-descent. We also show that the inclusion of the class of effective global-descent maps in the class surjective effective étale-descent is strict.Partial financial support by Centro de Matemática da Universidade de Coimbra is gratefully acknowledged.     

Towards Stone duality for topological theories

In the context of categorical topology, more precisely that of T-categories (Hofmann, 2007 [8]), we define the notion of T-colimit as a particular colimit in a V-category. A complete and cocomplete V-category in which limits distribute over T-colimits, is to be thought of as the generalisation of a (co-)frame to this categorical level. We explain some ideas on a T-categorical version of “Stone duality”, and show that Cauchy completeness of a T-category is precisely its sobriety.     

Filter spaces

The category FIL of filter spaces and cauchy maps is a topological universe. This paper establishes the foundation for a completion theory forT ₂ filter spaces.     

Compactness and subsets of ordered sets that meet all maximal chains

LetP be a chain complete ordered set. By considering subsets which meet all maximal chains, we describe conditions which imply that the space of maximal chains ofP is compact. The symbolsP ₁ andP ₂ refer to two particular ordered sets considered below. It is shown that the space of maximal chains (P) is compact ifP satisfies any of the following conditions: (i)P contains no copy ofP ₁ or its dual and all antichains inP are finite. (ii)P contains no properN and every element ofP belongs to a finite maximal antichain ofP. (iii)P contains no copy ofP ₁ orP ₂ and for everyx inP there is a finite subset ofP which is coinitial abovex. We also describe an example of an ordered set which is complete and densely ordered and in which no antichain meets every maximal chain.     

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.     

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.     

Closed structures on categories of topological spaces

The category of all topological spaces and continuous maps and its full subcategory of all T_o-spaces admit (up to isomorphism) precisely one structure of symmetric monoidal closed category (see [2]). In this paper we shall prove the same result for any epireflective subcategory of the category of topological spaces (particularly e.g. for the categories of Hausdorff spaces, regular spaces, Tychonoff spaces).     

Exponentiation for unitary structures

Motivated by the observation that both pretopologies and preapproach limits can be characterized as those convergence relations which have a unit for a suitable composition, we introduce the category Algu(T;V) of reflexive and unitary lax algebras, for a symmetric monoidal closed lattice V and a Set-monad T=(T,e,m). For T=U the ultrafilter monad, we characterize exponentiable morphisms in Algu(U;V). Further, we give a sufficient condition for an object to be exponentiable in the category Alg(U;V) of reflexive and transitive lax algebras. This specializes to known and new results for pretopological, preapproach and approach spaces.     

Ambiguous loci of the metric projection onto compact starshaped sets in a Banach space

LetE be a real Banach space andL(E) the family of all nonempty compact starshaped subsets ofE. Under the Hausdorff distance,L(E) is a complete metric space. The elements of the complement of a first Baire category subset ofL(E) are called typical elements ofL(E). ForXL(E) we denote by the metrical projection ontoX, i.e. the mapping which associates to eachaE the set of all points inX closest toa. In this note we prove that, ifE is strictly convex and separable with dimE2, then for a typicalXL(E) the map is not single valued at a dense set of points. Moreover, we show that a typical element ofL(E) has kernel consisting of one point and set of directions dense in the unit sphere ofE.