Some topologies on the spaces of USCO maps and densely continuous forms

We study the completeness of three (metrizable) uniformities on the sets D(X, Y) and U(X, Y) of densely continuous forms and USCO maps from X to Y: the uniformity of uniform convergence on bounded sets, the Hausdorff metric uniformity and the uniformity U _B . We also prove that if X is a nondiscrete space, then the Hausdorff metric on real-valued densely continuous forms D(X, ?) (identified with their graphs) is not complete. The key to guarantee completeness of closed subsets of D(X, Y) equipped with the Hausdorff metric is dense equicontinuity introduced by Hammer and McCoy in [7].     

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.     

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.     

Functional characterizations of p-spaces

We show that a completely regular space Y is a p-space (a ?ech-complete space, a locally compact space) if and only if given a dense subspace A of any topological space X and a continuous f: A → Y there are a p-embedded subset (resp. a G _δ-subset, an open subset) M of X containing A and a quasicontinuous subcontinuous extension f*: M → Y of f continuous at every point of A. A result concerning a continuous extension to a residual set is also given.     

Realcompactness of hyperspaces and extensions of open-closed maps

In response to questions of Ginsburg [9, 10], we prove that if cf(c)>ω₁, then there exists an open-closed, continuous map f from a normal, realcompact space X onto a space Y which is not realcompact. By his result the hyperspace 2^x of closed subsets of X is then not realcompact, and the extension μf(vf) of f to the topological completion (the Hewitt realcompactification) of X is not onto. The latter fact solves problems raised by Morita [16] and by Isiwata [12] both negatively. We also consider the problem whether or not the hyperspace of a hereditarily Lindelöf space is hereditarily realcompact.     

Functors on the category of quasi-fibrations

We consider the following questions: when can we extend a continuous endofunctor on Top the category of topological spaces to a fibrewise continuous endofunctor on Top(2) the category of continuous maps? If this is true, does such fibrewise continuous endofunctor preserve fibrations? In this paper, we define Fib the topological category of cell-wise trivial fibre spaces over polyhedra and show that any continuous endofunctor on Top induces a fibrewise continuous endofunctor on Fib preserving the class of quasi-fibrations.     

On chaotic extensions of dynamical systems

In this paper, inspired by some results in linear dynamics, we will show that every dynamical system (X,f), where f is a continuous self-map on a separable metric space X, can be extended to a chaotic (in the sense of Devaney) dynamical system in an isometric way.     

On von neumann's variation of the weierstrass-stone theorem

Let X be a compact HausdorfF space and let D(X) be the set of all continuous real-valued functions f defined on X and such that 0 ≤ f(x) ≤ 1, for all x ? X. The set D(X) is equipped with the uniform topology. We characterize the uniform closure of subsets A ? D(X) containing 0 and 1 and ?ψ + (1 ? ?)η, whenever they contain ?, ψ and η     

Devaney’s chaos on uniform limit maps

Let (X, d) be a compact metric space and f_n : X → X a sequence of continuous maps such that (f_n) converges uniformly to a map f. The purpose of this paper is to study the Devaney’s chaos on the uniform limit f. On the one hand, we show that f is not necessarily transitive even if all f_n mixing, and the sensitive dependence on initial conditions may not been inherited to f even if the iterates of the sequence have some uniform convergence, which correct two wrong claims in [1]. On the other hand, we give some equivalence conditions for the uniform limit f to be transitive and to have sensitive dependence on initial conditions. Moreover, we present an example to show that a non-transitive sequence may converge uniformly to a transitive map.     

On the existence of continuous (approximate) roots of algebraic equations

The present paper considers the existence of continuous roots of algebraic equations with coefficients being continuous functions defined on compact Hausdorff spaces. For a compact Hausdorff space X, C(X) denotes the Banach algebra of all continuous complex-valued functions on X with the sup norm ∥⋅_∥∞. The algebra C(X) is said to be algebraically closed if each monic algebraic equation with C(X) coefficients has a root in C(X). First we study a topological characterization of a first-countable compact (connected) Hausdorff space X such that C(X) is algebraically closed. The result has been obtained by Countryman Jr, Hatori-Miura and Miura-Niijima and we provide a simple proof for metrizable spaces.Also we consider continuous approximate roots of the equation zn−f=0 with respect to z, where f∈C(X), and provide a topological characterization of compact Hausdorff space X with dimX?1 such that the above equation has an approximate root in C(X) for each f∈C(X), in terms of the first ?ech cohomology of X.     

The average-shadowing property and strong ergodicity

Let X be a compact metric space and f:X→X be a continuous map. In this paper, we prove that if f has the average-shadowing property and the minimal points of f are dense in X, then f is weakly mixing and totally strongly ergodic. As applications we obtain that if f is a distal or Lyapunov stable map having the average-shadowing property, then X is consisting of one point. Moreover, we illustrate that the full shift has the average-shadowing property.     

Hereditary invertible linear surjections and splitting problems for selections

Let A+B be the pointwise (Minkowski) sum of two convex subsets A and B of a Banach space. Is it true that every continuous mapping h:X→A+B splits into a sum h=f+g of continuous mappings f:X→A and g:X→B? We study this question within a wider framework of splitting techniques of continuous selections. Existence of splittings is guaranteed by hereditary invertibility of linear surjections between Banach spaces. Some affirmative and negative results on such invertibility with respect to an appropriate class of convex compacta are presented. As a corollary, a positive answer to the above question is obtained for strictly convex finite-dimensional precompact spaces.     

Paths through inverse limits

In Bani?, ?repnjak, Merhar and Milutinovi? (2010) [2] the authors proved that if a sequence of graphs of surjective upper semi-continuous set-valued functions fn:X→X² converges to the graph of a continuous single-valued function f:X→X, then the sequence of corresponding inverse limits obtained from fn converges to the inverse limit obtained from f. In this paper a more general result is presented in which surjectivity of fn is not required. The result is also generalized to the case of inverse sequences with non-constant sequences of bonding maps. Finally, these new theorems are applied to inverse limits with tent maps. Among other applications, it is shown that the inverse limits appearing in the Ingram conjecture (with a point added) form an arc.     

Uniformly holomorphic continuation in locally convex spaces

We investigate the simultaneous uniformly holomorphic continuation of the uniformly holomorphic functions defined in a domain spread of uniform type, (X, ϑ), over a locally convex Hausdorff space E. We construct the envelope of uniform holomorphy of (X, ϑ) with an analogous method of the results of M. Schottenloher (Portugal. Math. 33 (1974)). Finally, we use this construction to the problem of extending uniformly holomorphic maps f: (X, ϑ) → F, with values in a complete locally convex space to the envelope of uniform holomorphy of X.     

Analytic functions whose range is dense in a ball

Denote by Δ(resp. $\text{Δ}$) the open (resp. closed) unit disc in C. Let E be a closed subset of the unit circle T and let F be a relatively closed subset of T ? E of Lesbesgue measure zero. The following result is proved. Given a complex Banach space X and a bounded continuous function f:F → X, there exists an extension f? of f, bounded and continuous on $\text{}\text{?}\text{gD ? E}$, analytic on Δ and satisfying sup$\text{{6}\text{f}\text{?}\text{(z)6:zε}\text{δ}\text{?E}$. This is applied to show that for any separable complex Banach space X there exists an analytic function from Δ to X whose range is contained and dense in the unit ball of X.     

Contractive maps on normed linear spaces and their applications to nonlinear matrix equations

In this paper we will give necessary and sufficient conditions under which a map is a contraction on a certain subset of a normed linear space. These conditions are already well known for maps on intervals in R. Using the conditions and Banach’s fixed point theorem we can prove a fixed point theorem for operators on a normed linear space. The fixed point theorem will be applied to the matrix equation X = I_n + A^∗f(X)A, where f is a map on the set of positive definite matrices induced by a real valued map on (0, ∞). This will give conditions on A and f under which the equation has a unique solution in a certain set. We will consider two examples of f in detail. In one example the application of the fixed point theorem is the first step in proving that the equation has a unique positive definite solution under the conditions on A.     

Duality and operator algebras: automatic weak* continuity and applications

We investigate some subtle and interesting phenomena in the duality theory of operator spaces and operator algebras, and give several applications of the surprising fact that certain maps are always weak^*-continuous on dual spaces. In particular, if X is a subspace of a C^*-algebra A, and if a∈A satisfies aX⊂X, then we show that the function x?ax on X is automatically weak^* continuous if either (a) X is a dual operator space, or (b) a^*X⊂X and X is a dual Banach space. These results hinge on a generalization to Banach modules of Tomiyama's famous theorem on contractive projections onto a C^*-subalgebra. Applications include a new characterization of the σ-weakly closed (possibly nonunital and nonselfadjoint) operator algebras, and a generalization of the theory of W^*-modules to the framework of modules over such algebras. We also give a Banach module characterization of σ-weakly closed spaces of operators which are invariant under the action of a von Neumann algebra.     

Commutative Abelian Galois extensions of a Banach algebra

Let A be a commutative unital Banach algebra with connected maximal ideal space X. We show that the Gelfand transform induces an isomorphism between the group of commutative Galois extensions of A with given finite Abelian Galois group, and the corresponding group of extensions of C(X). This result is applied, when X is sufficiently nice, to construct a separable projective finitely generated faithful Banach A-algebra whose maximal ideal space is a given finitely fibered covering space of X.     

Semigroups and abstract Gevrey spaces

Conditions are found under which a closed linear operator A in a Banach space X generates a continuous semigroup in a linear topological space Y which is dense in X. The space Y is an abstract Gevrey space associated with the operator A. This is an abstract setting for some results for hyperbolic systems with data in spaces of Gevrey functions.