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.     

TOPOLOGICAL CATEGORIES AND CLOSURE OPERATORS

Abstract

It is shown that the category CS of closure spaces is a topological category. For each epireflective subcategory A of a topological category X a functor F _A :X → X is defined and used to extend to the general case of topological categories some results given in [4], [5] and [10] for epireflective subcategories of the category Top of topological spaces.     

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.     

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).     

Function spaces and one point extensions for the construct of metered spaces

The construct M of metered spaces and contractions is known to be a superconstruct in which all metrically generated constructs can be fully embedded. We show that M has one point extensions and that quotients in M are productive. We construct a Cartesian closed topological extension of M and characterize the canonical function spaces with underlying sets Hom(X,Y) for metered spaces X and Y. Finally we obtain an internal characterization of the objects in the Cartesian closed topological hull of M.     

Conway’s Question: The Chase for Completeness

We study various degrees of completeness for a Tychonoff space X. One of them plays a central role, namely X is called a Conway space if X is sequentially closed in its Stone–?ech compactification β X (a prominent example of Conway spaces is provided by Dieudonné complete spaces). The Conway spaces constitute a bireflective subcategory Conw of the category Tych of Tychonoff spaces. Replacing sequential closure by the general notion of a closure operator C, we introduce analogously the subcategory Conw _C of C-Conway spaces, that turns out to be again a bireflective subcategory of Tych. We show that every bireflective subcategory of Tych can be presented in this way by building a Galois connection between bireflective subcategories of Tych and closure operators of Top finer than the Kuratowski closure. Other levels of completeness are considered for the (underlying topological spaces of) topological groups. A topological group G is sequentially complete if it is sequentially closed in its Ra?kov completion ${ \ifmmode\expandafter\tilde\else\expandafter\~\fi{G}}$ . The sequential completeness for topological groups is stronger than Conway’s property, although they coincide in some classes of topological groups, for example: free (Abelian) topological groups, pseudocompact groups, etc.     

The Cartesian product of a compactum and a space is a bifunctor in shape

In 2003 the author has associated with every cofinite inverse system of compact Hausdorff spaces X with limit X and every simplicial complex K (possibly infinite) with geometric realization P=|K| a resolution R(X,K) of X×P, which consists of paracompact spaces. If X consists of compact polyhedra, then R(X,K) consists of spaces having the homotopy type of polyhedra. In two subsequent papers the author proved that R(X,K) is a covariant functor in each of its variables X and K. In the present paper it is proved that R(X,K) is a bifunctor. Using this result, it is proved that the Cartesian product X×Z of a compact Hausdorff space X and a topological space Z is a bifunctor SSh(Cpt)×Sh(Top)→Sh(Top) from the product category of the strong shape category of compact Hausdorff spaces SSh(Cpt) and the shape category Sh(Top) of topological spaces to the category Sh(Top). This holds in spite of the fact that X×Z need not be a direct product in Sh(Top).     

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.     

Characterizing continuous functions on compact spaces

We consider the following problem: given a set X and a function , does there exist a compact Hausdorff topology on X which makes T continuous? We characterize such functions in terms of their orbit structure. Given the generality of the problem, the characterization turns out to be surprisingly simple and elegant. Amongst other results, we also characterize homeomorphisms on compact metric spaces.     

The enriched Vietoris monad on representable spaces

Employing a formal analogy between ordered sets and topological spaces, over the past years we have investigated a notion of cocompleteness for topological, approach and other kind of spaces. In this new context, the down-set monad becomes the filter monad, cocomplete ordered set translates to continuous lattice, distributivity means disconnectedness, and so on. Curiously, the dual(?) notion of completeness does not behave as the mirror image of the one of cocompleteness; and in this paper we have a closer look at complete spaces. In particular, we construct the “up-set monad” on representable spaces (in the sense of L. Nachbin for topological spaces, respectively C. Hermida for multicategories); we show that this monad is of Kock–Zöberlein type; we introduce and study a notion of weighted limit similar to the classical notion for enriched categories; and we describe the Kleisli category of our “up-set monad”. We emphasise that these generic categorical notions and results can be indeed connected to more “classical” topology: for topological spaces, the “up-set monad” becomes the lower Vietoris monad, and the statement “X   is totally cocomplete if and only if X^op$X^{\mathrm{op}}$ is totally complete” specialises to O. Wyler's characterisation of the algebras of the Vietoris monad on compact Hausdorff spaces as precisely the continuous lattices.     

Canonical subbase-compactness of topological products

For topological products the concept of canonical subbase-compactness is introduced, and the question analyzed under what conditions such products are canonically subbase-compact in ZF-set theory.Results: (1) Products of finite spaces are canonically subbase-compact iff AC(fin), the axiom of choice for finite sets, holds.(2) Products of n-element spaces are canonically subbase-compact iff AC(<n), the axiom of choice for sets with less than n elements, holds.(3) Products of compact spaces are canonically subbase-compact iff AC, the axiom of choice, holds.(4) All powers XI of a compact space X are canonically subbase compact iff X is a Loeb-space.These results imply that in ZF the implications

     

Functoriality of the standard resolution of the Cartesian product of a compactum and a polyhedron

In a previous paper the author has associated with every inverse system of compact Hausdorff spaces X with limit X and every simplicial complex K (possibly infinite) with geometric realization P=|K| a resolution RK(X) of X×P, which consists of paracompact spaces. If X consists of compact polyhedra, then RK(X) consists of spaces having the homotopy type of polyhedra. In the present paper it is proved that this construction is functorial. One of the consequences is the existence of a functor from the strong shape category of compact Hausdorff spaces X to the shape category of spaces, which maps X to the Cartesian product X×P. Another consequence is the theorem which asserts that, for compact Hausdorff spaces X, X^′, such that X is strong shape dominated by X^′ and the Cartesian product X^′×P is a direct product in Sh(Top), then also X×P is a direct product in the shape category Sh(Top).     

LOCAL COMPACTNESS IN SEMI-UNIFORM CONVERGENCE SPACES

Abstract

Local compactness is studied in the highly convenient setting of semi-uniform convergence spaces which form a common generalization of (symmetric) limit spaces (and thus of symmetric topological spaces) as well as of uniform limit spaces (and thus of uniform spaces). It turns out that it leads to a cartesian closed topological category and, in contrast to the situation for topological spaces, the local compact spaces are exactly the compactly generated spaces. Furthermore, a one-point Hausdorff compactification for noncompact locally compact Hausdorff convergence spaces is considered.¹     

An asymmetric Ellis theorem

In 1957 Robert Ellis proved that a group with a locally compact Hausdorff topology T making all translations continuous also has jointly continuous multiplication and continuous inversion, and is thus a topological group. The theorem does not apply to locally compact asymmetric spaces such as the reals with addition and the topology of upper open rays. We first show a bitopological Ellis theorem, and then introduce a generalization of locally compact Hausdorff, called locally skew compact, and a topological dual, Tk, to obtain the following asymmetric Ellis theorem which applies to the example above:Whenever (X,⋅,T) is a group with a locally skew compact topology making all translations continuous, then multiplication is jointly continuous in both (X,⋅,T) and (X,⋅,Tk), and inversion is a homeomorphism between (X,T) and (X,Tk).This generalizes the classical Ellis theorem, because T=Tk when (X,T) is locally compact Hausdorff.     

Functoriality of the standard resolution of the Cartesian product of a compactum and a polyhedron II

In 2003 the author has associated with every cofinite inverse system of compact Hausdorff spaces X with limit X and every simplicial complex K (possibly infinite) with geometric realization P=|K| a resolution R(X,K) of X×P, which consists of paracompact spaces. If X consists of compact polyhedra, then R(X,K) consists of spaces having the homotopy type of polyhedra. In a subsequent paper, published in 2007, the author proved that R(X,K) is a covariant functor in the first variable. In the present paper it is proved that R(X,K) is a covariant functor also in the second variable.     

Topological sums and products in ZF-set theory

In ZF, i.e., Zermelo-Fraenkel set theory without the axiom of choice, the category Top of topological spaces and continuous maps is well-behaved. In particular, Top has sums (=coproducts) and products. However, it may happen that for families (Xi)_i∈I and (Yi)_i∈I with the property that each Xi is homeomorphic to the corresponding Yi neither their sums _⊕i∈IXi and _⊕i∈IYi nor their products _∏i∈IXi and _∏i∈IYi are homeomorphic. It will be shown that the axiom of choice is not only sufficient but also necessary to rectify this defect.     

EPIREFLECTIVE SUBCATEGORIES AND SEMICLOSURE OPERATORS

ABSTRACT

Let C be a category of topological spaces and continuous functions which is full, hereditary and closed under homeomorphisms and products. If A is a subclass of C, let E(A) be the full subcategory of C whose objects are the subspaces in A. In this paper we characterize the epireflective subcategories of C containing A and contained in E(A) by introducing a “semiclosure” operator which is a generalization for the “idempotent semi-limit” operator introduced by S.S. Hong (see [5]) with respect to Top _o. In case A is extensive in C, so that E(A) = C, all the extensive subcategories of C containing A are thus characterized.