Open mappings looking like projections

Every open continuous mappingf from a metric space (X, d) onto a countable-dimensional metric spaceY admits a special type of factorization (Y×[0, 1] throughout), provided all fibers off are dense in itself and complete with respect tod. On this basis, an upper semi-continuous Cantor bouquet of disjoint usco selections for a class of 1.s.c. mappings between metrizable spaces is constructed.     

Some characterizations of spaces that have transfinite dimension

We characterize separable metrizable spaces that have small transfinite dimension and metrizable spaces that have large transfinite dimension modifying two classical characterizations of countable-dimensional spaces and applying the notion of a strongly point-finite family.     

Mappings onto metric spaces

We give characterizations of perfect images and open and compact images of spaces that can be mapped onto metrizable spaces by a mapping with fibers having a given property $\text{P}$. We use these characterizations to obtain conditions which imply that such images can be mapped onto a metric space by a mapping with fibers satisfying $\text{P}$. Such a treatment includes the investigation of spaces with a weaker metric topology [2, Ch. 5].     

Straight nearness spaces

Straight spaces are spaces for which a continuous map defined on the space which is uniformly continuous on each set of a finite closed cover is then uniformly continuous on the whole space. Previously, straight spaces have been studied in the setting of metric spaces. In this paper, we present a study of straight spaces in the more general setting of nearness spaces. In a subcategory of nearness spaces somewhat more general than uniform spaces, we relate straightness to uniform local connectedness. We investigate category theoretic situations involving straight spaces. We prove that straightness is preserved by final sinks, in particular by sums and by quotients, and also by completions.     

Cartesian-closed coreflective subcategories of uniform spaces generated by classes of metric spaces

The main result, in Theorem 3, is that in the category Unif of Hausdorff uniform spaces and uniformly continuous maps, the coreflective hulls of the following classes are cartesian-closed: all metric spaces having no infinite uniform partition, all connected metric spaces, all bounded metric spaces, and all injective metric spaces.Furthermore, Theorems 1 and 4 imply that if $\text{C}$ is any coreflective, cartesian-closed subcategory of Unif in which enough function space structures are finer than the uniformity of uniform convergence (as in the above examples), then either (1) $\text{C}$ is a subclass of the locally fine spaces, or (2) $\text{C}$ contains all injective metric spaces and $\text{C}$ is a subclass of the coreflective hull of all uniform spaces having no infinite uniform partition.     

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.     

Bead spaces and fixed point theorems

The notion of a bead metric space defined here (see Definition 6) is a nice generalization of that of the uniformly convex normed space. In turn, the idea of a central point for a mapping when combined with the “single central point” property of the bead spaces enables us to obtain strong and elegant extensions of the Browder-Göhde-Kirk fixed point theorem for nonexpansive mappings (see Theorems 14-17). Their proofs are based on a very simple reasoning. We also prove two theorems on continuous selections for metric and Hilbert spaces. They are followed by fixed point theorems of Schauder type. In the final part we obtain a result on nonempty intersection.     

Certain classes of weakly infinite-dimensional spaces and topological games

In [V.V. Fedorchuk, Questions on weakly infinite-dimensional spaces, in: E.M. Pearl (Ed.), Open Problems in Topology II, Elsevier, Amsterdam, 2007, pp. 637-645; V.V. Fedorchuk, Weakly infinite-dimensional spaces, Russian Math. Surveys 42 (2) (2007) 1-52] classes w-m-C of weakly infinite-dimensional spaces, 2?m?∞, were introduced. We prove that all of them coincide with the class wid of all weakly infinite-dimensional spaces in the Alexandroff sense. We show also that transfinite dimensions dimwm, introduced in [V.V. Fedorchuk, Questions on weakly infinite-dimensional spaces, in: E.M. Pearl (Ed.), Open Problems in Topology II, Elsevier, Amsterdam, 2007, pp. 637-645; V.V. Fedorchuk, Weakly infinite-dimensional spaces, Russian Math. Surveys 42 (2) (2007) 1-52], coincide with dimension dimw2=dim, where dim is the transfinite dimension invented by Borst [P. Borst, Classification of weakly infinite-dimensional spaces. I. A transfinite extension of the covering dimension, Fund. Math. 130 (1) (1988) 1-25]. Some topological games which are related to countable-dimensional spaces, to C-spaces, and some other subclasses of weakly infinite-dimensional spaces are discussed.     

Selection properties of uniform and related structures

In this paper we present some results on selection properties in asymmetric generalized metric and uniform spaces. We demonstrate differences between selection properties of these spaces and selection properties of metric and uniform spaces.     

On convergence in fuzzy metric spaces

The concept of p-convergence in fuzzy metric spaces, in George and Veeramani's sense, has been recently given by D. Mihet in [D. Mihet, On fuzzy contractive mappings in fuzzy metric spaces, Fuzzy Sets and Systems 158 (2007) 915-921]. In this note we study some aspects relative to this concept and characterize those fuzzy metric spaces, that we call principal, in which the family of p-convergent sequences agrees with the family of convergent sequences. Also a non-completable fuzzy metric space, which is not principal, is given.     

Monotone generalized contractions in partially ordered probabilistic metric spaces

In this paper, a concept of monotone generalized contraction in partially ordered probabilistic metric spaces is introduced and some fixed and common fixed point theorems are proved. Presented theorems extend the results in partially ordered metric spaces of Nieto and Rodriguez-Lopez [Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations, Order 22 (2005) 223-239; Existence and uniqueness of fixed point in partially ordered sets and applications to ordinary differential equations, Acta Math. Sin. (Engl. Ser.) 23 (2007) 2205-2212], Ran and Reurings [A fixed point theorem in partially ordered sets and some applications to matrix equations, Proc. Amer. Math. Soc. 132 (2004) 1435-1443] to a more general class of contractive type mappings in partially ordered probabilistic metric spaces and include several recent developments.     

Global inversion and covering maps on length spaces

In order to obtain global inversion theorems for mappings between length metric spaces, we investigate sufficient conditions for a local homeomorphism to be a covering map in this context. We also provide an estimate of the domain of invertibility of a local homeomorphism around a point, in terms of a kind of lower scalar derivative. As a consequence, we obtain an invertibility result using an analog of the Hadamard integral condition in the frame of length spaces. Some applications are given to the case of local diffeomorphisms between Banach-Finsler manifolds. Finally, we derive a global inversion theorem for mappings between stratified groups.     

On metric spaces and local extrema

The main aim of this paper is to give a positive answer to a question of Behrends, Geschke and Natkaniec regarding the existence of a connected metric space and a non-constant real-valued continuous function on it for which every point is a local extremum. Moreover we show that real-valued continuous functions on connected spaces such that every family of pairwise disjoint non-empty open sets is of size <|R| are constant provided that every point is a local extremum.     

Amenability,locally finite spaces,and bi-lipschitz embeddings

We define the isoperimetric constant for any locally finite metric space and we study the property of having isoperimetric constant equal to zero. This property, called Small Neighborhood property, clearly extends amenability to any locally finite space. Therefore, we start making a comparison between this property and other notions of amenability for locally finite metric spaces that have been proposed by Gromov, Lafontaine and Pansu, by Ceccherini-Silberstein, Grigorchuk and de la Harpe and by Block and Weinberger. We discuss possible applications of the property SN in the study of embedding a metric space into another one. In particular, we propose three results: we prove that a certain class of metric graphs that are isometrically embeddable into Hilbert spaces must have the property SN. We also show, by a simple example, that this result is not true replacing property SN with amenability. As a second result, we prove that many spaces with uniform bounded geometry having a bi-lipschitz embedding into Euclidean spaces must have the property SN. Finally, we prove a Bourgain-like theorem for metric trees: a metric tree with uniform bounded geometry and without property SN does not have bi-lipschitz embeddings into finite-dimensional Hilbert spaces.     

On dimension of inverse limits with upper semicontinuous set-valued bonding functions

We give results about the dimension of continua, obtained by combining inverse limits of inverse sequences of metric spaces and one-valued bonding maps with inverse limits of inverse sequences of metric spaces and upper semicontinuous set-valued bonding functions, by standard procedure introduced in [I. Bani?, Continua with kernels, Houston J. Math. (2006), in press].     

Between compactness and completeness

Call a sequence in a metric space cofinally Cauchy if for each positive ε there exists a cofinal (rather than residual) set of indices whose corresponding terms are ε-close. We give a number of new characterizations of metric spaces for which each cofinally Cauchy sequence has a cluster point. For example, a space has such a metric if and only each continuous function defined on it is uniformly locally bounded. A number of results exploit a measure of local compactness functional that we introduce. We conclude with a short proof of Romaguera's Theorem: a metrizable space admits such a metric if and only if its set of points having a compact neighborhood has compact complement.     

Coupled coincidence point theorems under contractive conditions in partially ordered probabilistic metric spaces

In this paper, we introduce the concept of a mixed g-monotone mapping and prove coupled coincidence and coupled common fixed point theorems under ?-contractive conditions for self-maps in partially ordered complete probabilistic metric spaces.     

Infinitesimal characterization of a feedback nash equilibrium in a differential game

For mappings defined on metric spaces and with values in Banach spaces, the notions of derivative vectors of first and second order are introduced. These notions are used to establish both necessary and sufficient optimality conditions of first and second order for local ?-minimizers of such mappings, where ? is a strict preorder relation defined on the space of values of the mapping that is minimized. As corollaries of the above results, minimality conditions are also obtained for the case when the mapping is defined on a subset of a normed space.     

Endpoints of set-valued dynamical systems of asymptotic contractions of Meir–Keeler type and strict contractions in uniform spaces

In this paper, we introduce the concepts of the set-valued dynamical systems of asymptotic contractions of Meir–Keeler type and set-valued dynamical systems of strict contractions in uniform spaces and we present a method which is useful for establishing conditions guaranteeing the existence and uniqueness of endpoints of these contractions and the convergence to these endpoints of all generalized sequences of iterations of these contractions. The result, concerning the investigations of problems of the set-valued asymptotic fixed point theory, include some well-known results of Meir and Keeler, Kirk and Suzuki concerning the asymptotic fixed point theory of single-valued maps in metric spaces. The result, concerning set-valued strict contractions (in which the contractive coefficient is not constant), is different from the result of Yuan concerning the existence of endpoints of Tarafdar–Vyborny generalized contractions (in which the contractive coefficient is constant) in bounded metric spaces and provides some examples of Tarafdar–Yuan topological contractions in compact uniform spaces. Definitions and results presented here are new for set-valued dynamical systems in uniform, locally convex and metric spaces and even for single-valued maps. Examples show a fundamental difference between our results and the well-known ones.     

On countable-to-one maps

In this paper, it is proved that a space with a point-countable base is an open, countable-to-one image of a metric space, and a quotient, countable-to-one image of a metric space is characterized by a point-countable ℵ₀-weak base. Examples are provided in order to answer negatively questions posed by Gruenhage et al. [G. Gruenhage, E. Michael, Y. Tanaka, Spaces determined by point-countable covers, Pacific J. Math. 113 (1984) 303-332] and Tanaka [Y. Tanaka, Closed maps and symmetric spaces, Questions Answers Gen. Topology 11 (1993) 215-233].