Inverse limits of arcs and of simple closed curves with confluent bonding mappings

It is proved that Knaster's type continua and solenoids can be considered as inverse limits of arcs and of circles with confluent bonding mappings. Several other classes of bonding mappings, which are relative to confluent ones, also are discussed.     

Open retractions onto n-ods

We characterize dendrites which admit an open retraction onto an arbitrary n-od. The paper contains also several corollaries and two problems concerning the subject     

Selections for quasi-l.s.c. mappings with uniformly equi-LCn range

Every quasi-lower semi-continuous (q.l.s.c.) mapping admits a lower semi-continuous (l.s.c.) selection preserving all important (from the selection point of view) properties of the former mapping. Special-type extensions of l.s.c. mappings are established on this base.     

Completely positive mappings and mean matrices

Some functions f:R⁺→R⁺ induce mean of positive numbers and the matrix monotonicity gives a possibility for means of positive definite matrices. Moreover, such a function f can define a linear mapping on matrices (which is basic in the constructions of monotone metrics). The present subject is to check the complete positivity of in the case of a few concrete functions f. This problem has been motivated by applications in quantum information.     

On a generalization of perfectly normal spaces

The notion of quasi-perfect normality which generalizes the notion of usual perfect normality is introduced and the behavior of the dimension functions in this class of spaces is investigated.     

Exactly (n, 1) mappings on graphs

We characterize those graphs which admit exactly (n, 1) mappings onto themselves. We also suggest two questions concerning the subject.     

Open selections on smooth fans

Given a dendroid X, an open selection is an open map such that s(A)∈A for every A∈C(X). We show that a smooth fan X admits an open selection if and only if X is locally connected.     

On projections and limit mappings of inverse systems of compact spaces

A compactificaton αX of a completely regular space X is “determined” by a subset F of C¹(X) if αX is the smallest compactificaton of X to which each element of F extends, and is “generated” by F if the evaluation map $\text{e}{}_{\mathrm{F}}\text{:X →}\text{R}{}^{\mathrm{n}}\text{,n = |F|}$, is an embedding and $\text{αX =}\text{e}{}_{\mathrm{F}}\text{(X)}$. Evidently, if F either determines or generates αX, then every elements of F has an extension to αX; whenever F satisfies this latter condition, the set of all such extensions is denoted F^α.A major results of our previous paper is that F determines αX if and only if F^α separates points of αX ? X. A major result of the present paper is that F generates αX if and only if F^α separates points of αX.     

REAL COMPACTIFICATIONS THROUGH ZERO-SET SPACES

Abstract

The Alexandroff (= zero-set) spaces were introduced in [l] as the “completely normal spaces”, and have been studied in a number of more recent papers. In this paper we unify the theory of Wallman realcompactifications via the Alexandroff bases and introduce the realcompactfine Alexandroff spaces as particularly relevant to their investigation. These latter spaces are defined analogously to the A-c uniform spaces which are based on a construction of A.W. Hager [25].     

Quadrature rule-based bounds for functions of adjacency matrices

Bounds for entries of matrix functions based on Gauss-type quadrature rules are applied to adjacency matrices associated with graphs. This technique allows to develop inexpensive and accurate upper and lower bounds for certain quantities (Estrada index, subgraph centrality, communicability) that describe properties of networks.     

Multi-valued contraction mappings in metric spaces

Summary Some fixed point theorems for multi-valued contraction mappings in metric spaces, related to a fixed point theorem due to T. Zamfirescu [6], [5], are presented.     

Generic continuity of restricted weak upper semi-continuous set-valued mappings

Set-valued mappings from a topological space into subsets of a Banach space which satisfy a restricted form of weak upper semi-continuity, have particularly noteworthy properties. We establish a selection theorem for certain set-valued mappings from a (-) unfavourable topological space into subsets of a Banach space and as a consequence derive the property that restricted weak upper semi-continuous set-valued mappings which satisfy a minimality condition, from a (-) unfavourable topological space into subsets of a Banach space are single-valued and norm upper semi-continuous at the points of a residual subset of their domain.     

Local selections and local dendrites

Let X be a Peano continuum, C(X) its space of subcontinua, and C(X, ε) the space of subcontinua of diameter less than ε. A selection on some subspace of C(X) is a continuous choice function; the selection σ is rigid if σ(A) ? B ? A implies σ(A) = σ(B). It is shown that X is a local dendrite (contains at most one simple closed curve) if and only if there exists ε > 0 such that C(X, ε) admits a selection (rigid selection). Further, C(X) admits a local selection at the subcontinuum A if and only if A has a neighborhood (relative to the space C(X)) which contains no cyclic local dendrite; moreover, that local selection may be chosen to be a constant.     

Extending paired multifunctions

As a rule, the classical Michael-type selection theorems for the existence of single-valued selections are analogues and, in certain respects, generalisations of ordinary extension theorems. In contrast to this, the theorems for the existence of multi-selections deal with natural generalisations of cover properties of topological spaces. This paper continues the study of the latter problem, and its main purpose is to furnish a mapping characterisation of a cover-extension property—the so-called Katětov spaces.     

Selections for paraconvex-valued mappings on non-paracompact domains

We prove that Michael?s paraconvex-valued selection theorem for paracompact spaces remains true for C^′(E)-valued mappings defined on collectionwise normal spaces. Some possible generalisations are also given.     

Local compactness and Hewitt realcompactifications of products II

We generalize and refine results from the author's paper [18]. For a completely regular Hausdorff space X, υX denotes the Hewitt realcompactification of X. It is proved that if υ(X×Y)=υX×υY for any metacompact subparacompact (or m-paracompact) space Y, then X is locally compact. A P(n)-space is a space in which every intersection of less than n open sets is open. A characterization of those spaces X such that υ (X×Y = υX×υY for any (metacompact) P(n)-space Y is also obtained.     

Some representations of midconvex set-valued functions

Summary In this note we establish conditions under which every midconvex set-valued function can be represented as sum of an additive function and a convex set-valued function. These results improve some theorems obtained in [8], [10] and [3]. Some results on local Jensen selections of midconvex set-valued functions are also given.     

The orderability of nonarchimedean spaces

Let X be a nonarchimedean space and C be the union of all compact open subsets of X. The following conditions are listed in increasing order of generality. (Conditions 2 and 3 are equivalent.) 1. X is perfect; 2. C is an F_σ in X; 3. C? is metrizable; 4. X is orderable. It is also shown that X is orderable if $\text{C}\text{?}\text{?C}$ is scattered or X is a GO space with countably many pseudogaps. An example is given of a non-orderable, totally disconnected, GO space with just one pseudogap.     

Fixed point theorems for partially outward mappings

We generalize some classical theorems related to dimension. We extend Brouwer's fixed point theorem to a class of mappings whose images are not necessarily a subset of the domain. These results also generalize theorems of B.R. Halpern and G.M. Bergman. As applications, we prove some theorems for maps that pull absolute retracts outward into attached sphere collars. We note relationships to the relative Nielsen theory and show that certain of our applications can also be obtained using results of H. Schirmer.