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).     

BIFRAMES AND BISPACES

Abstract

The concept of a biframe is introduced. Then the known dual adjunction between topological spaces and frames (i.e. local lattices) is extended to one between bispaces (i.e. bitopological spaces) and biframes. The largest duality contained in this dual adjunction defines the sober bispaces, which are also characterized in terms of the sober spaces. The topological and the frame-theoretic concepts of regularity, complete regularity and compactness are extended to bispaces and biframes respectively. For the bispaces these concepts are found to coincide with those introduced earlier by J.C. Kelly, E.P. Lane, S. Salbany and others. The Stone-?ech compactification (compact regular coreflection) of a biframe is constructed without the Axiom of Choice.     

On the tychonoff functor and w-compactness

The main purpose of this paper is to settle the following problem concerning a product formula for the Tychonoff functor τ, by introducing the notion of w-compact spaces: Characterize a topological space X such that τ(X×Y)=τ(X)×τ(Y) for any topological space Y. We also study the properties of w-compact spaces, and it is proved that, for any family {X_α} of w-compact spaces, the product ΠX_α is also w-compact and τ(ΠX_α)=Πτ(X_α).     

Collectionwise Hausdorff versus collectionwise normal with respect to compact sets

Collectionwise normal (CWN) and collectionwise Hausdorff (CWH) spaces have played an increasingly important role in topology since the introduction of these concepts by R.H. Bing in 1951 [3]. It has remained an open and frequently raised question as to whether CWH T₃-spaces are CWN with respect to compact sets. Recently, a counterexample requiring the existence of measurable cardinals and having little additional topological structure was constructed by W.G. Fleissner and the author. In this paper, the author gives a simple example in ZFC of a CWH, first countable, perfect T₃-space that is not CWN with respect to compact, metrizable sets, and, under Martin's Axiom, such an example that is also a Moore space. In addition, the author considers the analogous question for strongly collectionwise Hausdorff (SCWH) T₃-spaces and characterizes the existence of SCWH T₃-spaces that are not CWN with respect to compact sets in set-theoretic and box product formulations. The constructions utilized throughout the paper are of a general nature and several apparently new set-theoretic techniques for interchanging ‘points’ and ‘sets’ are introduced.     

Real dicompactifications of ditopological texture spaces

Real dicompactifications and dicompactifications of a ditopological texture space are defined and studied.Section 2 considers nearly plain extensions of a ditopological texture space (S,S,τ,κ). Spaces that possess a nearly plain extension are shown to have a property, called here almost plainness, that is weaker than that of near plainness, but which shares with near plainness the existence of an associated plain space (Sp,Sp,τp,κp). Some properties of the class of almost plain ditopological texture spaces are established, a notion of canonical nearly plain extension of an almost plain ditopological texture space, projective and injective pre-orderings and the concept of isomorphism on such canonical nearly plain extensions are defined.In Section 3 the notion of nearly plain extension is specialized to that of real dicompactification and dicompactification, and the spaces that have such extensions are characterized. Working in terms of a specific representation of the canonical real dicompactifications and dicompactifications of a completely biregular bi-T₂ almost plain ditopological space, the interrelation between sub-T-lattices of the T-lattice of ω-preserving bicontinuous real mappings on the associated plain space and the real dicompactifications and dicompactifications are investigated. In particular generalizations of the Hewitt realcompactification and Stone-?ech compactification are obtained, and shown to be reflectors for the appropriate categories.     

Some generalizations of Fedorchuk duality theorem—I

Generalizing duality theorem of V.V. Fedorchuk [V.V. Fedorchuk, Boolean δ-algebras and quasi-open mappings, Sibirsk. Mat. Zh. 14 (5) (1973) 1088-1099; English translation: Siberian Math. J. 14 (1973) 759-767 (1974)], we prove Stone-type duality theorems for the following four categories: the objects of all of them are the locally compact Hausdorff spaces, and their morphisms are, respectively, the continuous skeletal maps, the quasi-open perfect maps, the open maps, the open perfect maps. In particular, a Stone-type duality theorem for the category of compact Hausdorff spaces and open maps is obtained. Some equivalence theorems for these four categories are stated as well; two of them generalize the Fedorchuk equivalence theorem [V.V. Fedorchuk, Boolean δ-algebras and quasi-open mappings, Sibirsk. Mat. Zh. 14 (5) (1973) 1088-1099; English translation: Siberian Math. J. 14 (1973) 759-767 (1974)].     

Real dicompact textures

A notion of real compactness for completely biregular bi-T₂ ditopological texture spaces is defined and studied under the name real dicompactness. In particular it is shown that real dicompact spaces are nearly plain ∗-spaces, and an important characterization is presented. Finally the connection of this work with topological and bitopological real compactness is discussed in a categorical setting.     

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.     

Infinite distributive laws versus local connectedness and compactness properties

Various local connectedness and compactness properties of topological spaces are characterized by higher degrees of distributivity for their lattices of open (or closed) sets, and conversely. For example, those topological spaces for which not only the lattice of open sets but also that of closed sets is a frame, are described by the existence of web neighborhood bases, where webs are certain specific path-connected sets. Such spaces are called web spaces. The even better linked wide web spaces are characterized by F-distributivity of their topologies, and the worldwide web spaces (or C-spaces) by complete distributivity of their topologies. Similarly, strongly locally connected spaces and locally hypercompact spaces are characterized by suitable infinite distributive laws. The web space concepts are also viewed as natural extensions of spaces that are semilattices with respect to the specialization order and have continuous (unary, binary or infinitary) semilattice operations.     

Topo-canonical completions of closure algebras and Heyting algebras

We introduce and investigate topo-canonical completions of closure algebras and Heyting algebras. We develop a duality theory that is an alternative to Esakia’s duality, describe duals of topo-canonical completions in terms of the Salbany and Banaschewski compactifications, and characterize topo-canonical varieties of closure algebras and Heyting algebras. Consequently, we show that ideal completions preserve no identities of Heyting algebras. We also characterize definable classes of topological spaces. Received January 20, 2006; accepted in final form September 12, 2006.     

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.     

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.     

Every topological category is convenient for Gelfand duality

In this paper we generalize our work on Gelfand dualities in cartesian closed topological categories [42] to categories which are only monoidally closed. Using heavily enriched category theory we show that under very mild conditions on the base category function algebra functor and spectral space functor exist, forming a pair of adjoint functors and establishing a duality between function algebras and spectral spaces. Using recent results in connection with semitopological functors, we show that every (E,M)-topological category is endowed with at least oneconvenient monoidal structure admitting a generalized Gelfand duality. So it turns out that there is no need for a cartesian closed structure on a topological category in order to study generalized Gelfand-Naimark dualities.     

Amalgams, connectifications, and homogeneous compacta

We construct a path-connected homogeneous compactum with cellularity c that is not homeomorphic to any product of dyadic compacta and first countable compacta. We also prove some closure properties for classes of spaces defined by various connectifiability conditions. One application is that every infinite product of infinite topological sums of Ti spaces has a Ti pathwise connectification, where i∈{1,2,3,3.5}.     

CONCRETE CATEGORIES FOR WHICH FIBRE COMPLETIONS INDUCE TOPOLOGICAL COMPLETIONS

Abstract

Every topological category over an arbitrary base category X may be considered as a category of T-models with respect to some theory (i.e., functor) T from X into a category of complete lattices. Using this model-theoretic correspondence as our basic tool, we study initial and final completions of (co)fibration complete categories. For an arbitrary concrete category (A, U) over X, the process of order-theoretically completing each fibre does not usually yield an initial/final completion of (A, U). It is shown in this paper that for concrete categories which are assumed to be fibration and/or cofibration complete, initial and final completions can be constructed by completing the fibres. These completions are further shown to exhibit some interesting external properties.     

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.     

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