Almost isometric embedding between metric spaces

We investigate the relations ofalmost isometric embedding and ofalmost isometry between metric spaces. These relations have several appealing features. For example, all isomorphism types of countable dense subsets of ∝ form exactly one almostisometry class, and similarly with countable dense subsets of Uryson's universal separable metric spaceU. We investigate geometric, set-theoretic and model-theoretic aspects of almost isometry and of almost isometric embedding. The main results show that almost isometric embeddability behaves in the category ofseparable metric spaces differently than in the category of general metric spaces. While in the category of general metric spaces the behavior of universality resembles that in the category of linear orderings —namely, no universal structure can exist on a regular λ > ℵ₁ below the continuum—in the category of separable metric spaces universality behaves more like that in the category of graphs, that is, a small number of metric separable metric spaces on an uncountable regular λ<2^ℵ ₀ may consistently almost isometrically embed all separable metric spaces on λ. Research of the first author was supported by an Israeli Science foundation grant no. 177/01. Research of the second author was supported by the United States-Israel Binational Science Foundation. Publication 827.     

S-types of Global Towers of Spaces and Exterior Spaces

The closed model category of exterior spaces, that contains the proper category, is a useful tool for the study of non compact spaces and manifolds. The notion of exterior weak ℕ-S-equivalences is given by exterior maps which induce isomorphisms on the k-th ℕ-exterior homotopy groups for k ∈ S, where S is a set of non negative integers. The category of exterior spaces with a base ray localized by exterior weak ℕ-S-equivalences is called the category of exterior ℕ-S-types. The existence of closed model structures in the category of exterior spaces permits to establish equivalences between homotopy categories obtained by dividing by exterior homotopy relations, and categories of fractions (localized categories) given by the inversion of classes of week equivalences. The family of neighbourhoods ‘at infinity’ of an exterior space can be interpreted as a global prospace and under the condition of first countable at infinity we can consider a global tower instead of a prospace. The objective of this paper is to use localized categories to find the connection between S-types of exterior spaces and S-types of global towers of spaces. The main result of this paper establishes an equivalence between the category of S-types of rayed first countable exterior spaces and the category of S-types of global towers of pointed spaces. As a consequence of this result, categories of global towers of algebraic models localized up to weak equivalences can be used to give some algebraic models of S-types. The authors acknowledge the financial support given by the projects FOMENTA 2007/03 and MTM2007-65431.     

Cochains and homotopy type

Finite type nilpotent spaces are weakly equivalent if and only if their singular cochains are quasi-isomorphic as E_∞ algebras. The cochain functor from the homotopy category of finite type nilpotent spaces to the homotopy category of E_∞ algebras is faithful but not full.     

Intuitionistic <Emphasis Type="Italic">I</Emphasis>-fuzzy topological spaces

The main purpose of this paper is to introduce the concept of intuitionistic I-fuzzy quasi-coincident neighborhood systems of intuitiostic fuzzy points. The relation between the category of intuitionistic I-fuzzy topological spaces and the category of intuitionistic I-fuzzy quasi-coincident neighborhood spaces are studied. By using fuzzifying topology, the notion of generated intuitionistic I-fuzzy topology is proposed, and the connections among generated intuitionistic I-fuzzy topological spaces, fuzzifying topological spaces and I-fuzzy topological spaces are discussed. Finally, the properties of the operators Iω, ι are obtained.     

Invariants of Directed Spaces

Directed spaces are the objects of study within directed algebraic topology. They are characterised by spaces of directed paths associated to a source and a target, both elements of an underlying topological space. The algebraic topology of these path spaces and their connections are studied from a categorical perspective. In particular, we study the preorder category associated to a directed space and various “quotient” categories arising from algebraic topological functors. Furthermore, we propose and study a new notion of directed homotopy equivalence between directed spaces.      

A de Vries-type duality theorem for the category of locally compact spaces and continuous maps. I

Generalizing de Vries’ duality theorem [9], we prove that the category HLC of locally compact Hausdorff spaces and continuous maps is dual to the category DHLC of complete local contact algebras and appropriate morphisms between them.     

On some properties of bitopological QHC spaces

Some important questions connected with bitopological QHC spaces are investigated. New conditions are found, under which such spaces are compact with respect to one component of the topology. It is shown that a pairwise extremal disconnected bitopological QHC space is S-closed in the sense of [4]. Theorems on the second category of a base set and on the almost Baire property of bitopological QHC spaces are proved. Also, several properties of QHC bitopological spaces are found under some known bitopological mappings. Published in Lietuvos Matematikos Rinkinys, Vol. 46, No. 2, pp. 186–192, April–June, 2006.     

Deformation-obstruction theory for complexes via Atiyah and Kodaira–Spencer classes

We give a universal approach to the deformation-obstruction theory of objects of the derived category of coherent sheaves over a smooth projective family. We recover and generalise the obstruction class of Lowen and Lieblich, and prove that it is a product of Atiyah and Kodaira–Spencer classes. This allows us to obtain deformation-invariant virtual cycles on moduli spaces of objects of the derived category on threefolds.     

Relative Riemann-Zariski spaces

In this paper we study relative Riemann-Zariski spaces associated to a morphism of schemes and generalizing the classical Riemann-Zariski space of a field. We prove that similarly to the classical RZ spaces, the relative ones can be described either as projective limits of schemes in the category of locally ringed spaces or as certain spaces of valuations. We apply these spaces to prove the following two new results: a strong version of stable modification theorem for relative curves; a decomposition theorem which asserts that any separated morphism between quasi-compact and quasiseparated schemes factors as a composition of an affine morphism and a proper morphism. In particular, we obtain a new proof of Nagata’s compactification theorem.     

THE FORGETFUL FUNCTOR Cont → Prox IS TOPOLOGICAL

Abstract

It is shown that the forgetful functor from the category of contiguity spaces to the category of generalized proximity spaces is topological, and that the right adjoint right inverse of this functor extends the inverse of the forgetful functor from the category of totally bounded uniform spaces to the category of proximity spaces.     

Functorial polar functions

W _∞ denotes the category of archimedean ℓ-groups with designated weak unit and complete ℓ-homomorphisms that preserve the weak unit. CmpT _2,∞ denotes the category of compact Hausdorff spaces with continuous skeletal maps. This work introduces the concept of a functorial polar function on W _∞ and its dual a functorial covering function on CmpT _2,∞.     

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.     

A note on recursive completions of the countable atomless Boolean algebras

The Baire category theorem for compact Hausdorff spaces is used to show that any recursive completion ℬ^ϕof the countable atomless Boolean algebra ℬ is ℬrecursively closed." Also, a theorem of R.D. Anderson on the simplicity of Aut(ℬ) is used to show that uncountably many automorphisms of ℬ extend to ℬ^ϕ. This revised version was published online in June 2006 with corrections to the Cover Date.     

Strong inclusion orders between L-subsets and its applications in L-convex spaces

Abstract

In this paper, the concept of strong inclusion orders between L-subsets is introduced. As a tool, it is applied to the following aspects. Firstly, the notion of algebraic L-closure operators is proposed and the resulting category is shown to be isomorphic to the category of L-convex spaces (also called algebraic L-closure spaces). Secondly, restricted L-hull operators, as generalizations of restricted hull operators, are introduced and the resulting category is also proved to be isomorphic to the category of L-convex spaces. Finally, by using the properties of strong inclusion orders, it is shown that the category of convex spaces can be embedded in the category of stratified L-convex spaces as a reflective subcategory and the concrete form of the coreflective functor from the category of L-convex spaces to the category of stratified L-convex spaces is presented.     

Abstract ordered compact convex sets and algebras of the (sub)probabilistic powerdomain monad over ordered compact spaces

The majority of categories used in denotational semantics are topological in nature. One of these is the category of stably compact spaces and continuous maps. Previously, Eilenberg–Moore algebras were studied for the extended probabilistic powerdomain monad over the category of ordered compact spaces X and order-preserving continuous maps in the sense of Nachbin. Appropriate algebras were characterized as compact convex subsets of ordered locally convex topological vector spaces. In so doing, functional analytic tools were involved. The main accomplishments of this paper are as follows: the result mentioned is re-proved and is extended to the subprobabilistic case; topological methods are developed which defy an appeal to functional analysis; a more topological approach might be useful for the stably compact case; algebras of the (sub)probabilistic powerdomain monad inherit barycentric operations that satisfy the same equational laws as those in vector spaces. Also, it is shown that it is convenient first to embed these abstract convex sets in abstract cones, which are simpler to work with. Lastly, we state embedding theorems for abstract ordered locally compact cones and compact convex sets in ordered topological vector spaces.     

Morse�CBott functions and the Lusternik�CSchnirelmann category

Lusternik–Schnirelmann category of a manifold gives a lower bound of the number of critical points of a differentiable map on it. The purpose of this paper is to show how to construct cone-decompositions of manifolds by using functions of class C ¹ and their gradient flows, where cone-decompositions are used to give an upper bound for the Lusternik–Schnirelmann category which is a homotopy invariant of a topological space. In particular, the Morse–Bott functions on the Stiefel manifolds considered by Frankel (1965) are effectively used to construct the conedecompositions of Stiefel manifolds and symmetric Riemannian spaces to determine their Lusternik–Schnirelmann categories.     

Symmetric Hilbert spaces arising from species of structures

Symmetric Hilbert spaces such as the bosonic and the fermionic Fock spaces over some lsquo;one particle space’ are formed by certain symmetrization procedures performed on the full Fock space. We investigate alternative ways of symmetrization by building on Joyal's notion of a combinatorial species. Any such species F gives rise to an endofunctor of the category of Hilbert spaces with contractions mapping a Hilbert space to a symmetric Hilbert space with the same symmetry as the species F. A general framework for annihilation and creation operators on these spaces is developed, and compared to the generalised Brownian motions of R. Speicher and M. Bożejko. As a corollary we find that the commutation relation with admits a realization on a symmetric Hilbert space whenever f has a power series with infinite radius of convergence and positive coefficients. Received: 7 April 2000; in final form: 28 November 2000 / Published online: 19 October 2001     

Quantified functional analysis and seminormed spaces: A dual adjunction

In this work we investigate the natural algebraic structure that arises on dual spaces in the context of quantified functional analysis. We show that the category of absolutely convex modules is obtained as the category of Eilenberg-Moore algebras induced by the dualization functor [−,R] on locally convex approach spaces. We also establish a dual adjunction between the latter category and the category of seminormed spaces.     

EXPONENTIAL LAW AND θ-CONTINUOUS FUNCTIONS

Abstract

The category θ-Top of topological spaces and θ-continuous functions is not Cartesian closed; but it is known that under certain local property assumptions, the exponential law in θ-Top is fulfilled. We define a functor from θ-Top to the category of H-θ-topological spaces and prove that in this category the exponential law holds without any local property assumptions. We also provide a functor from θ-Top to Katětov's category of filter-merotopic spaces, which is Cartesian closed.     

The Weizsäcker phenomenon and a canonical definition of Gaussian Lebesgue–Rokhlin measures

The notion of “Gaussian Lebesgue–Rokhlin space” is introduced. The definition is canonical, i.e., is given without use of topological and others irrelevant mathematical structures. The object under discussion completes the category of finite-dimensional Gaussian vector spaces. Some nontrivial examples are considered and historical comments are given. Bibliography: 30 titles.