Janusz J. Charatonik

Monotone mappings of universal dendrites

Abstract For each m ∈{3,4,…,Ω} mappings of the standard universaldendrite D m of order m onto itself are studied which belong to the following classes: homeomorphisms, near homeomorphisms and monotone mappings. In particular, it is shown that each such dendrite D m is homogenous with respect to monotone mappings. The obtained results extend ones due to H. Kato.

MONOTONE MAPPINGS AND UNICOHERENCE AT SUBCONTINUA

Abstract It is shown that unicoherence at subcontinua is preserved under a new class of mappings between metric continua which comprises the class of monotone and the class of hereditarily confluent mappings, while it is not preserved under open finite-to-one mappings or under quasi-monotone mappings even between linear graphs.

On projections and limit mappings of inverse systems of compact spaces

Abstract For some classes K of mappings we discuss two problems connected with limits of inverse systems: (1) Does the condition that all bonding mappings are in K imply that all projections are in K ? (2) Does the condition that all mappings between factor spaces of two given inverse systems are in K imply that the limit mapping between the inverse limit spaces is in K ? We answer both these questions in the affirmative for the classes of monotone, of confluent and of weakly confluent mappings of compact spaces, and for some generalizations of these mappings.

On covering mappings on solenoids

Covering mappings on the dyadic solenoid are studied. Some results stated by Zhou Youcheng (2000) are discussed in a more detailed way by indicating certain inaccuracies in their proofs. These are either corrected or supplemented, or else suitable counterexamples are constructed. Some open questions are asked and connections with related results are considered.

Fans with the property of Kelley

Abstract Fans having the property of Kelley are characterized as the limits of inverse sequences of finite fans with confluent bonding mappings and as smooth fans for which the set of their end-points together with the top is closed. Also a characterization is obtained of smooth fans with the set of end-points closed as the limits of inverse sequences of finite fans with open bonding mappings.

On feebly monotone and related classes of mappings

Abstract A mapping between continua is said to be feebly monotone if whenever the range is the union of two proper subcontinua, their preimages are connected. Basic properties of these mappings and their connections with related classes of mappings are investigated. Further, some special properties of continua as indecomposability, irreducibility, unicoherence, and some other are studied when applied to either the domain or the range of the considered mapping. Finally terminal subcontinua and related concepts are discussed pertinent to feebly monotone mappings.

Openness of induced mappings

Abstract Openness of induced mappings between hyperspaces of continua is studied. In particular we investigate continua X such that if for a mapping f:X→Y the induced mapping C(f):C(X)→C(Y) is open, then f is a homeomorphism. It is shown that, besides hereditarily locally connected continua, all fans have this property, while some Cartesian products do not have it. If f:X×Y→X denotes the natural projection, then openness of C(f) implies that X is hereditarily unicoherent. The equivalence holds for dendrites. Some new characterizations of these curves are obtained.

History of Continuum Theory

By a continuum we usually mean a metric (or Hausdorff) compact connected space. The original definition of 1883, due to Georg Cantor, [126], p. 576, stated that a subset of a Euclidean space is a continuum provided it is perfect (i.e. closed and dense-in-itself, or — equivalently — coincides with its first derivative) and connected, i.e. if for every two of its points a and b and for each positive number є there corresponds a finite system of points a = p 0, p 1, ..., p n = b such that the distance between any two consecutive points of the system is less than є. The equivalence of the two definitions for compact metric spaces is shown e.g. in Kuratowski’s monograph [390], vol. 2, §47, I, Theorem 0, p. 167.

MAPPINGS OF THE SIERPIÑSKI CURVE ONTO ITSELF

Given two points p and q of the Sierpinski universal plane curve S, necessary and/or sufficient conditions are discussed in the paper under which there is a mapping f of S onto itself such that f(p) = q and / be- longs to one of the following: homeomorphisms, local homeomorphisms, local homeomorphisms in the large sense, open, simple or monotone mappings. 1. Introduction. The paper is devoted to the problem of homogeneity of the Sierpinski universal plane curve from one point to another with respect to various classes of continuous mappings. The Krasinkiewicz result for homeomorphisms (4) is extended to local homeomorphisms and also the problem is completely solved for local homeomorphisms in the large sense. It is also shown that the Sierpinski curve is homogeneous with respect to simple mappings and with respect to mono- tone ones. Furthermore, the Whyburn result (10) on an extension of a homeo- morphism between boundaries of two complementary domains to one between the whole Sierpinski curves is generalized to open mappings. Some unresolved problems are posed in the final part of the paper. Thanks are due to W. J. Charatonik and J. Nikiel for fruitful discussions on the topic of this paper. 2. Preliminaries. All mappings considered in the paper are assumed to be continuous. A curve means a one-dimensional metric continuum. By the standard Sierpinski curve we mean the well-known geometric realization of a plane locally connected curve (see e.g. (6, §51, I, Example 5, p. 275 and Figure 8, p. 276)) which is located in the unit square I2 with opposite vertices (0,0) and (1,1), and which is known to be universal in the class of all plane curves (see e.g. (1, Theorem 12.11, p. 433)). Any homeomorphic image of this continuum is called the Sierpinski curve, and is denoted by S. The union of all boundaries of complementary domains of S in the plane is called the rational part of S and is denoted by R. The remaining part, S\R, of the curve is called its irrational part (cf. (3, p. 188; 4, p. 255)). Let a class M of mappings be given. A space X is said to be homogeneous with respect to M from a point p E X to a point q E X provided there exists a mapping / of X onto itself such that f E M and f(p) = q. This is a generalization of a concept of a space being homogeneous between points p and q (see (4, p. 255)). A space X is said to be homogeneous with respect to M provided that it is homogeneous with respect to M from p to q for each pair of points p,q E X. This generalizes the concept of a homogeneous space (i.e. homogeneous with respect to homeomorphisms).

Limit properties of induced mappings

Abstract Given a class M of mappings f between continua, near- M stands for the class of uniform limits of sequences of mappings from M . Let 2 f and C(f) mean the induced mappings between hyperspaces. Relations are studied between the conditions: f∈ near- M , 2 f ∈ near- M and C(f)∈ near- M . A special attention is paid to the classes M of open and of monotone mappings.