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.     

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.     

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.     

Bornologies and metrically generated theories

Bornologies axiomatize an abstract notion of bounded sets and are introduced as collections of subsets satisfying a number of consistency properties. Bornological spaces form a topological construct, the morphisms of which are those functions which preserve bounded sets. A typical example is a bornology generated by a metric, i.e. the collection of all bounded sets for that metric. In a recent paper [E. Colebunders, R. Lowen, Metrically generated theories, Proc. Amer. Math. Soc. 133 (2005) 1547-1556] the authors noted that many examples are known of natural functors describing the transition from categories of metric spaces to the “metrizable” objects in some given topological construct such that, in some natural way, the metrizable objects generate the whole construct. These constructs can be axiomatically described and are called metrically generated. The construct of bornological spaces is not metrically generated, but an important large subconstruct is. We also encounter other important examples of metrically generated constructs, the constructs of Lipschitz spaces, of uniform spaces and of completely regular spaces. In this paper, the unified setting of metrically generated theories is used to study the functorial relationship between these constructs and the one of bornological spaces.     

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.     

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

     

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).     

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.     

A theorem of Hrushovski-Solecki-Vershik applied to uniform and coarse embeddings of the Urysohn metric space

A theorem proved by Hrushovski for graphs and extended by Solecki and Vershik (independently from each other) to metric spaces leads to a stronger version of ultrahomogeneity of the infinite random graph R, the universal Urysohn metric space U, and other related objects. We show how the result can be used to average out uniform and coarse embeddings of U (and its various counterparts) into normed spaces. Sometimes this leads to new embeddings of the same kind that are metric transforms and besides extend to affine representations of various isometry groups. As an application of this technique, we show that U admits neither a uniform nor a coarse embedding into a uniformly convex Banach space.     

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).     

Uniformities with the same Hausdorff hypertopology

Two uniformities U and V on a set X are said to be H-equivalent if their corresponding Hausdorff uniformities on the set of all non-empty subsets of X induce the same topology. The uniformity U is said to be H-singular if no distinct uniformity on X is H-equivalent to U. The self-explanatory concepts of H-coarse, H-minimal and H-maximal uniformities are defined similarly.It is well known that not all uniformities are H-singular. We show here that there is a property which obstructs H-singularity: Every H-minimal uniformity has a base of finite-dimensional uniform coverings. Besides, we provide an intrinsic characterization of H-minimal uniformities and show that they are H-coarse. This characterization of H-minimality becomes a criterion for H-singularity for all uniformities that are either complete, uniformly locally precompact or proximally fine (e.g., metrizable ones). Some relevant properties which insure H-singularity are introduced and investigated in some aspect.     

Moscow spaces and selection theory

A well-known result on Moscow spaces states that every Gδ-dense subset of a Moscow space X is C-embedded in X. We present here the selection version of this result and also (by means of two different approaches) we use selection theory to characterize the open bounded subsets of a uniform space (X,U) in the case when its completion is a Moscow space.     

Metrical characterization of super-reflexivity and linear type of Banach spaces

We prove that a Banach space X is not super-reflexive if and only if the hyperbolic infinite tree embeds metrically into X. We improve one implication of J.Bourgain’s result who gave a metrical characterization of super-reflexivity in Banach spaces in terms of uniform embeddings of the finite trees. A characterization of the linear type for Banach spaces is given using the embedding of the infinite tree equipped with the metrics d _p induced by the ℓ _p norms. Received: 2 August 2006, Revised: 10 April 2007     

Local connectedness and extension of uniformly continuous functions

A metric space X is straight if for each finite cover of X by closed sets, and for each real valued function f on X, if f is uniformly continuous on each set of the cover, then f is uniformly continuous on the whole of X. The straight spaces have been studied in [A. Berarducci, D. Dikranjan, J. Pelant, An additivity theorem for uniformly continuous functions, Topology and its Applications 146-147 (2005) 339-352], which contains characterization of the straight spaces within the class of the locally connected spaces (they are the uniformly locally connected ones) and the class of the totally disconnected spaces (they coincide with the totally disconnected Atsuji spaces). We show that the completion of a straight space is straight and we characterize the dense straight subspaces of a straight space. In order to clarify further the relation between straightness and the level of local connectedness of the space we introduce two more intermediate properties between straightness and uniform local connectedness and we give various examples to distinguish them. One of these properties coincides with straightness for complete spaces and provides in this way a useful characterization of complete straight spaces in terms of the behaviour of the quasi-components of the space.     

Stratifiable spaces defined by pair collections

We define a pair (F,U) to be a closed set F and an open set U such that F ? U. A sequence of pair collections is used to characterize stratifiable spaces instead of a sequence of neighbornets. We introduce a new class of spaces, called regularly stratifiable spaces, which is defined in terms of pair collections. Every stratifiable μ -space is regularly stratifiable, and every regularly stratifiable space has a σ -almost locally finite base, thus is hereditary M₁. J. Nagata's problem for the dimension of M₁ -spaces is answered positively in the class of regularly stratifiable spaces.     

The bicompletion of the Hausdorff quasi-uniformity

We study conditions under which the Hausdorff quasi-uniformity UH of a quasi-uniform space (X,U) on the set P₀(X) of the nonempty subsets of X is bicomplete.Indeed we present an explicit method to construct the bicompletion of the T₀-quotient of the Hausdorff quasi-uniformity of a quasi-uniform space. It is used to find a characterization of those quasi-uniform T₀-spaces (X,U) for which the Hausdorff quasi-uniformity of their bicompletion on is bicomplete.     

Infima of quasi-uniform anti-atoms

Let (q(X),⊆) denote the lattice consisting of the set q(X) of all quasi-uniformities on a set X, ordered by set-theoretic inclusion ⊆. We observe that a quasi-uniformity on X is the supremum of atoms of (q(X),⊆) if and only if it is totally bounded and transitive. Each quasi-uniformity on X that is totally bounded or has a linearly ordered base is shown to be the infimum of anti-atoms of (q(X),⊆). Furthermore, each quasi-uniformity U on X such that the topology of the associated supremum uniformity Us is resolvable has the latter property.