PROXIMAL LIMIT SPACES

Abstract

The proximal limit spaces are introduced which fill the gap arising from the existence of proximity spaces, uniform spaces, and uniform limit spaces. It is shown that the proximal limit spaces can be considered as a bireflective subcategory of the topological category of uniform limit spaces. A limit space is induced by a proximal limit space if and only if it is a S₁-limit space.     

Spaces in which every dense subset is a Gδ

A topological space X is called a DG_δ-space if every subset of X is a G_δ-set in its closure. In this paper we study DG_δ-spaces that contains subspaces in which every dense subset is open and spaces in which every subset is a G_δ . We give some new results in these classes of topological spaces.     

PRODUCTS IN TOPOLOGY

Abstract

Examples are provided which demonstrate that in many cases topological products do not behave as they should. A new product for topological spaces is defined in a natural way by means of interior covers. In general this is no longer a topological space but can be interpreted as categorical product in a category larger than Top. For compact spaces the new product coincides with the old. There is a converse: For symmetric topological spaces X the following conditions are equivalent: (1) X is compact; (2) for each cardinal k the old and the new product X^k coincide; (3) for each compact Hausdorff space Y the old and the new product X x Y coincide. The new product preserves paracompactness, zero-dimensionality (in the covering sense), the Lindelöf property, and regular-closedness. With respect to the new product, a space is N-complete iff it is zerodimensional and R-complete.     

On the characterization of radical and semisimple classes in categories

We define and characterize radical and semisimple classes in a category K which satisfies certain conditions. These conditions are such that K could be any of the categories of associative rings, groupsR-modules, topological spaces or graphs. Among others, the following is proved:.

A class of objects R in K is a radical class if and only if K is a cohereditary component class which is closed under extensions and with T ? R. A class of objects S in K is a semisimple class if and only if S is a hereditary class which is closed under subdirect embed-dings and extensions with T ? S.     

Function Spaces and Dieudonné Completeness

Abstract

For a completely regular space X and a normed space E let C_k (x, E) (resp., C_p (x, E)) be the set of all E-valued continuous maps on X endowed with the compact-open (resp., pointwise convergence) topology. It is shown that the set of all F-valued linear continuous maps on C_k (x, E) when equipped with the topology of uniform convergence on the members of some families of bounded subsets of C_k (x, E) is a complete uniform space if F is a Band space and X is Dieudonné complete. This result is applied to prove that Dieudonné completeness is preserved by linear quotient surjections from C_k (x, E) onto C_k (Y, E) (resp., from C_p (x, E) onto C_p (x, E)) provided E, F are Band spaces and Y is a k-space.     

SUPERCATEGORIES OF THE CATEGORY OF NEARNESS SPACES

Abstract

The framework in which nearness spaces were defined by H. Herrlich [1] and [2], leads one to consider the supercategory Pow of the category Near of nearness spaces, having as objects all pairs (X,ξ), where X is a set and ξ ? P(P(X)) is any subset of the power set of the power set of X, and as morphisms f: (X,ξ) → (Y,n) all functions f: X → Y such that, if A ? ξ then fA □ {f(A) | A ξ A} ? η. In this paper we show that the full subcategories of Pow comprising the objects satisfying subsets of the prenearness space axioms lie in a lattice of bireflections or bicoreflections. This serves as a first step towards the aim of characterizing all bireflective (resp. bicoreflective) and even all initially complete subcategories of Pow.     

SYNTOPOFORMIC SPACES (SYNTOPOGENOUS LIMIT SPACES)

Abstract

Császár generalized the uniform spaces, the proximity spaces and the topological spaces to syntopogenous spaces. Cook and Fischer generalized the uniform spaces to uniform limit spaces. Finally Marny generalized the proximity spaces to proximal limit spaces. Analogously we generalize the syntopogenous spaces to syntopoformic spaces (syntopogenous limit spaces). These spaces include all the above mentioned in a suitable sense. We extend some of the well-known results of compactness and completeness to syntopoformic spaces.     

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.     

The Namioka property of KC-functions and Kempisty spaces

A topological space Y is called a Kempisty space if for any Baire space X every function , which is quasi-continuous in the first variable and continuous in the second variable has the Namioka property. Properties of compact Kempisty spaces are studied in this paper. In particular, it is shown that any Valdivia compact is a Kempisty space and the Cartesian product of an arbitrary family of compact Kempisty spaces is a Kempisty space.     

A remark on uniform spaces with invariant nonstandard hulls

Let (X , Γ) be a uniform space with its uniformity generated by a set of pseudo‐metrics Γ. Let the symbol ? denote the usual infinitesimal relation on *X , and define a new infinitesimal relation ≈ on *X by writing x ≈ y whenever *? (x, p ) ? *? (y, p ) for each ? ∈ Γ and each p ∈ X . We call (X , Γ) an S‐space if the relations ? and ≈ coincide on fin(*X ). S ‐spaces are interesting because their nonstandard hulls have representations within Nelson's internal set theory (IST, [5]). This was shown in [1], where it was also observed that the class of uniform spaces that have invariant nonstandard hulls is contained in the class of S ‐spaces. The question of whether there are S ‐spaces that do not have invariant nonstandard hulls was left open in [1]. In this note we show that when the uniformity of an S ‐space is given by a single pseudometric, the space has invariant nonstandard hulls. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

SMOOTHNESS PROPERTIES OF REAL-VALUED FUNCTIONS BASED ON THEIR OSCILLATION

Abstract

We consider the following two selection principles for topological spaces:

Principle 1: For each sequence of dense subsets, there is a sequence of points from the space, the n-th point coming from the n-th dense set, such that this set of points is dense in the space;

Principle 2: For each sequence of dense subsets, there is a sequence of finite sets, the n-th a subset of the n-th dense set, such that the union of these finite sets is dense in the space.

We show that for separable metric space X one of these principles holds for the space C_p (X) of realvalued continuous functions equipped with the pointwise convergence topology if, and only if, a corresponding principle holds for a special family of open covers of X. An example is given to show that these equivalences do not hold in general for Tychonoff spaces. It is further shown that these two principles give characterizations for two popular cardinal numbers, and that these two principles are intimately related to an infinite game that was studied by Berner and Juhász.     

On the duality between varieties and algebraic theories

It is proved that for any ultrametric space (X, d), the set L(X) of its closed balls is a lattice . It is complete, atomic, tree-like, and real graduated. For any such lattice , the set A(L) of its atoms can be naturally equipped with an ultrametric . These assignments are inverse of one another: where the first equality means an isometry while the second one is a lattice isomorphism. A similar correspondence established for morphisms, shows that there is an isomorphism of categories. The category ULTRAMETR of ultrametric spaces and non-expanding maps is isomorphic to the category LAT* of complete, atomic, tree-like, real graduated lattices and isotonic, semi-continuous, non-extensive maps. We describe properties of the isomorphism functor and its relations to the categorical operations and action of other functors. Basic properties of a space (such as completeness, spherical completeness, total boundedness, compactness, etc.) are translated into algebraic properties of the corresponding lattice L(X).     

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.     

On countably uniform closed-spaces

Abstract

Let X be a topological space and C_c(X) be the functionally countable subalgbera of C(X). We call X to be a countably uniform closed-space, briefly, a CU C-space, if C_c(X) is closed under uniform convergence. We investigate that countably uniform closedness need not closed under finite intersection and infinite product. It is shown that if X is a countable union of quasi-components, then X is a CU C-space. We characterize C_c-embedding and also -embedding in CU C-spaces. A subset S of X is called Z_c-embedded, if each Z ∈ Z_c(S) is the restriction of a zero-set of Z_c(X). It is observed that in a zero-dimensional CU C-space, each Lindelöf subspae is Z_c-embedded. Moreover, it is shown that in CU C-spaces, each Lindelöf subspace is C_c-embedded if and only if it is c-completely separated from each zero-set, which is disjoint from it. Also in latter spaces, it is observed that for each S ? X, C_c-embedding, -embedding and Z_c-embedding coincide, when S belongs to Z_c(X) or it is a c-pseudocompact space. Finally, when X is both a CU C-space and a CP-space, then each Z_c-embedded subspace is C_c-embedded (-embedded) in X.     

An extension of the dual complexity space and an application to Computer Science

In 1999, Romaguera and Schellekens introduced the theory of dual complexity spaces as a part of the development of a mathematical (topological) foundation for the complexity analysis of programs and algorithms [S. Romaguera, M.P. Schellekens, Quasi-metric properties of complexity spaces, Topology Appl. 98 (1999) 311-322]. In this work we extend the theory of dual complexity spaces to the case that the complexity functions are valued on an ordered normed monoid. We show that the complexity space of an ordered normed monoid inherits the ordered normed structure. Moreover, the order structure allows us to prove some topological and quasi-metric properties of the new dual complexity spaces. In particular, we show that these complexity spaces are, under certain conditions, Hausdorff and satisfy a kind of completeness. Finally, we develop a connection of our new approach with Interval Analysis.     

Dissipated compacta

The dissipated spaces form a class of compacta which contains both the scattered compacta and the compact LOTSes (linearly ordered topological spaces), and a number of theorems true for these latter two classes are true more generally for the dissipated spaces. For example, every regular Borel measure on a dissipated space is separable.The standard Fedor?uk S-space (constructed under ?) is dissipated. A dissipated compact L-space exists iff there is a Suslin line.A product of two compact LOTSes is usually not dissipated, but it may satisfy a weakening of that property. In fact, the degree of dissipation of a space can be used to distinguish topologically a product of n LOTSes from a product of m LOTSes.     

Injective spaces via adjunction

The work of the present author and his coauthors over the past years gives evidence that it may be useful to regard each topological space as a kind of enriched category, by interpreting the convergence relation x→x between ultrafilters and points of a topological space X as arrows in X. Naturally, this point of view opens the door to the use of concepts and ideas from enriched Category Theory for the investigation of topological spaces. Topological theories introduced by the author provide a convenient general setting for appropriately transferring these concepts and ideas to the world of topological spaces and some other geometric objects such as approach spaces. Using tools like adjunction and the Yoneda lemma, we show that the cocomplete spaces are precisely the injective spaces, and they are algebras for a suitable monad on . This way we obtain enriched versions of known results about injective topological spaces and continuous lattices.     

A metric characterization of the irrationals via a group operation

Let S be a separable metric space with a compatible metric d that satisfies: For each point x ? S and each nonnegative real number r there exists a unique point y ? S such that d(x,y) = r.In this paper spaces that meet the above criterion are investigated. It is shown that, under the assumption of completeness, this metric property characterizes the space of irrationals.     

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 results on separate and joint continuity

Let f:X×K→R be a separately continuous function and C a countable collection of subsets of K. Following a result of Calbrix and Troallic, there is a residual set of points x∈X such that f is jointly continuous at each point of {x}×Q, where Q is the set of y∈K for which the collection C includes a basis of neighborhoods in K. The particular case when the factor K is second countable was recently extended by Moors and Kenderov to any ?ech-complete Lindelöf space K and Lindelöf α-favorable X, improving a generalization of Namioka's theorem obtained by Talagrand. Moors proved the same result when K is a Lindelöf p-space and X is conditionally σ-α-favorable space. Here we add new results of this sort when the factor X is σC(X)-β-defavorable and when the assumption “base of neighborhoods” in Calbrix-Troallic's result is replaced by a type of countable completeness. The paper also provides further information about the class of Namioka spaces.