Pointfree Aspects of the Td Axiom of Classical Topology

Abstract

Abstraction from the condition defining To-spaces leads to the following notion in an arbitrary frame L: a filter F in L is called slicing if it is prime and there exist a, b e L such that a £ F, b e F, and a is covered by 6. This paper deals with various aspects of these slicing filters. As a first step, we present several results about the original td condition. Next, concerning slicing filters, we show they are completely prime and characterize them in various ways. In addition, we prove for the frames £>X of open subsets of a space X that every slicing filter is an open neighbourhood filter U(x) and X is td iff every U(x) is slicing. Further, for Top_D and Prm_D the categories of td spaces and their continuous maps, and all frames and those homomorphisms whose associated spectral maps preserve the completely prime elements, respectively, we show that the usual contravariant functors between Top and Frm induce analogous functors here, providing a dual equivalence between Top_D and the subcategory of Prm_D given by the To-spatial frames (not coinciding with the spatial ones). In addition, we show that Top_D is mono-coreflective in a suitable subcategory of Top. Finally, we provide a comparison between Jo-separation and sobriety showing they may be viewed, in some sense, as mirror images of each other.     

Locales as spectral spaces

A frame is a complete distributive lattice that satisfies the infinite distributive law ${b \wedge \bigvee_{i \in I} a_i = \bigvee_{i \in I} b \wedge a_i}$ b ∧ ? i ∈ I a i = ? i ∈ I b ∧ a i . The lattice of open sets of a topological space is a frame. The frames form a category Fr. The category of locales is the opposite category Fr ^op . The category BDLat of bounded distributive lattices contains Fr as a subcategory. The category BDLat is anti-equivalent to the category of spectral spaces, Spec (via Stone duality). There is a subcategory of Spec that corresponds to the subcategory Fr under the anti-equivalence. The objects of this subcategory are called locales, the morphisms are the localic maps; the category is denoted by Loc. Thus locales are spectral spaces. The category Loc is equivalent to the category Fr ^op . A topological approach to locales is initiated via the systematic study of locales as spectral spaces. The first task is to characterize the objects and the morphisms of the category Spec that belong to the subcategory Loc. The relationship between the categories Top (topological spaces), Spec and Loc is studied. The notions of localic subspaces and localic points of a locale are introduced and studied. The localic subspaces of a locale X form an inverse frame, which is anti-isomorphic to the assembly associated with the frame of open and quasi-compact subsets of X.     

Cartesian Closed Topological Categories and Tensor Products

The projective tensor product in a category of topological R-modules (where R is a topological ring) can be defined in Top, the category of topological spaces, by the same universal property used to define the tensor product of R-modules in Set. In this article, we extend this definition to an arbitrary topological category X and study how the Cartesian closedness of X is related to the monoidal closedness of the category of R-module objects in X. Mathematics Subject Classifications (2000) 18D15, 18D35, 18A40.     

Idempotency of Extensions via the Bicompletion

Let Top ₀ be the category of topological T ₀-spaces, QU ₀ the category of quasi-uniform T ₀-spaces, T : QU ₀ → Top ₀ the usual forgetful functor and K : QU ₀ → QU ₀ the bicompletion reflector with unit k : 1 → K. Any T-section F : Top ₀ → QU ₀ is called K-true if KF = FTKF, and upper (lower) K-true if KF is finer (coarser) than FTKF. The literature considers important T-sections F that enjoy all three, or just one, or none of these properties. It is known that T(K,k)F is well-pointed if and only if F is upper K-true. We prove the surprising fact that T(K,k)F is the reflection to Fix(TkF) whenever it is idempotent. We also prove a new characterization of upper K-trueness. We construct examples to set apart some natural cases. In particular we present an upper K-true F for which T(K,k)F is not idempotent, and a K-true F for which the coarsest associated T-preserving coreflector in QU ₀ is not stable under K. We dedicate this paper to the memory of Sérgio de Ornelas Salbany (1941–2005).     

The Absolute of a Topological Space and its Application to Abelian l-groups

We give a axiomatic treatment of the absolute in the category of all topological spaces and characterize it as a cover with respect to the full subcategory of extremally disconnected spaces. As an application, we obtain the strongly projectable hull of an abelian l-group as a unique lifting of the absolute of its spectrum. Dedicated to B. V. M.     

Bicompletion and Samuel Bicompactification

It is proved that the quasi-proximity space induced by the bicompletion of a quasi-uniform T ₀-space X is a subspace of the quasi-proximity space induced by the Samuel bicompactification of X. The result is then used to establish that the locally finite covering quasi-uniformity defined on the category Top ₀ of topological T ₀-spaces and continuous maps is not lower K-true (in the sense of Brümmer). It is also shown that a functorial quasi-uniformity F on Top ₀ is upper K-true if and only if FX is bicomplete whenever X is sober.     

Heredity and Coreflective Subcategories of the Category of Topological Spaces

Every hereditary coreflective subcategory of Top containing the category of finitely-generated spaces is shown to be generated by a class of spaces having a unique accumulation point. It is also shown that the coreflective hull of a union of two hereditary coreflective subcategories of Top need not be hereditary so that a coreflective subcategory of Top need not have a hereditary coreflective kernel.     

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.     

An Elementary Approach to Exponentiable Maps

Originally, exponentiable maps in the category Top of topological spaces were described by Niefield in terms of certain fibrewise Scott-open sets. This generalizes the first characterization of exponentiable spaces by Day and Kelly, which was improved thereafter by Hofmann and Lawson who described them as core-compact spaces.Besides various categorical methods, the Sierpinski-space is an essential tool in Niefield's original proof. Therefore, this approach fails to apply to quotient reflective subcategories of Top like Haus, the category of Hausdorff spaces. A recent generalization of the Hofmann–Lawson improvement to exponentiable maps enables now to reprove the characterization in a completely different and very elementary way. This approach works for any nontrivial quotient reflective subcategory of Top or Top/ _T , the category of all spaces over a fixed base space T, as well as for exponentiable monomorphisms with respect to epi-reflective subcategories.An important special case is the category Sep_Top/ _T of separated maps, i.e. distinct points in the same fibre can be separated in the total space by disjoint open neighbourhoods. The exponentiable objects in Sep turn out to be the open and fibrewise locally compact maps. The same holds for Haus/ _T , T a Hausdorff space. In this case, a similar characterization was obtained by Cagliari and Mantovani.     

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.     

Strong pro-fibrations and ANR objects in pro-categories

The notions of pro-fibration and approximate pro-fibration for morphisms in the pro-category pro-Top of topological spaces were introduced by S. Mardeši? and T.B. Rushing. In this paper we introduce the notion of strong pro-fibration, which is a pro-fibration with some additional property, and the notion of ANR object in pro-Top, which is approximately an ANR-system, and we consider the full subcategory ANR of pro-Top whose objects are ANR objects. We prove that the category ANR satisfies most of the axioms for fibration category in the sense of H.J. Baues if fibrations are strong pro-fibrations and weak equivalences are morphisms inducing isomorphisms in the pro-homotopy category pro-H(Top) of topological spaces. We give various applications. First of all, we prove that every shape morphism is represented by a strong pro-fibration. Secondly, the fibre of a strong pro-fibration is well defined in the category ANR, and we obtain an isomorphism between the pro-homotopy groups of the base and total systems of a strong pro-fibration, and hence obtain the pro-homotopy sequence of a strong pro-fibration. Finally, we also show that there is a homotopy decomposition in the category ANR.     

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.     

Constructions of Solid Hulls

For each concrete category (K,U) an extension LIM(K,U) is constructed and under certain 'smallness conditions' it is proved that LIM(K,U) is a solid hull of (K,U), i.e., the least finally dense solid extension of (K,U). A full subcategory of Top ₂ is presented which does not have a solid hull.     

On Hereditary Coreflective Subcategories of Top

Let A be a topological space which is not finitely generated and CH(A) denote the coreflective hull of A in Top. We construct a generator of the coreflective subcategory SCH(A) consisting of all subspaces of spaces from CH(A) which is a prime space and has the same cardinality as A. We also show that if A and B are coreflective subcategories of Top such that the hereditary coreflective kernel of each of them is the subcategory FG of all finitely generated spaces, then the hereditary coreflective kernel of their join CH(AB) is again FG.     

Normal subobjects of topological groups and of topological semi-Abelian algebras

As any category Gp(E) of internal groups in a given category E, the category Gp(Top) of topological groups possesses the strong algebraic property of protomodularity which carries intrinsic notions of normal subobject and of centrality. Here we explicit and investigate these intrinsic notions in the category Gp(Top). We extend these results to any category TopT of topological semi-Abelian algebras.     

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.     

L-fuzzy Q-convergence structures

In this paper, we construct a topological category of pretopological L-fuzzy Q-convergence spaces, which contains the category of topological L-fuzzy Q-convergence spaces as a bireflective full subcategory. Considering the connections with L-fuzzy topology, it is proved that the category of topological L-fuzzy Q-convergence spaces is isomorphic to the category of topological L-fuzzy quasi-coincident neighborhood spaces, and the latter is isomorphic to the category of L-fuzzy topological spaces. Moreover, we find that our pretopological L-fuzzy Q-convergence spaces can be characterized as a kind of L-fuzzy quasi-coincident neighborhood spaces, which is called strong L-fuzzy quasi-coincident neighborhood space.     

T0-SEPARATION IN TOPOLOGICAL CATEGORIES

R.-E. Hoffmann [5,6] has introduced the notion of an (E,M)-universally topological functor, which provides a categorical characterization of the T₀-separation axiom of general topology. In this paper, we characterise these functors in terms of the unique extension of structure functors defined on the subcategory of “separated” objects (of the domain category). This, in turn, leads to a solution of some problems due to G.C.L. Brümmer [1,2]. Other results include a generalization of L. Skula's characterization of the bireflective subcategories of Top [10].     

Functors on the category of quasi-fibrations

We consider the following questions: when can we extend a continuous endofunctor on Top the category of topological spaces to a fibrewise continuous endofunctor on Top(2) the category of continuous maps? If this is true, does such fibrewise continuous endofunctor preserve fibrations? In this paper, we define Fib the topological category of cell-wise trivial fibre spaces over polyhedra and show that any continuous endofunctor on Top induces a fibrewise continuous endofunctor on Fib preserving the class of quasi-fibrations.     

Hereditary,Additive and Divisible Classes in Epireflective Subcategories of Top

Hereditary coreflective subcategories of an epireflective subcategory A of Top such that I ₂ ? A (here I ₂ is the two-point indiscrete space) were studied in [4]. It was shown that a coreflective subcategory B of A is hereditary (closed under the formation subspaces) if and only if it is closed under the formation of prime factors. The main problem studied in this paper is the question whether this claim remains true if we study the (more general) subcategories of A which are closed under topological sums and quotients in A instead of the coreflective subcategories of A. We show that this is true if A ? Haus or under some reasonable conditions on B. E.g., this holds if B contains either a prime space, or a space which is not locally connected, or a totally disconnected space or a non-discrete Hausdorff space. We touch also other questions related to such subclasses of A. We introduce a method extending the results from the case of non-bireflective subcategories (which was studied in [4]) to arbitrary epireflective subcategories of Top. We also prove some new facts about the lattice of coreflective subcategories of Top and ZD.