Search citation statements
Paper Sections
Citation Types
Year Published
Publication Types
Relationship
Authors
Journals
Abstract.A continuum M is in class W provided that for each continuum Y and mapping f of Y onto M, each subcontinuum of M is the image under / of some subcontinuum of Y. It is shown that atriodic continua with trivial first Cech cohomology are in class W. Introduction.In 1972 Lelek introduced a class of continua known as class W. A continuum belongs to class W provided that each mapping from a continuum onto it is weakly confluent (for definition, see below). Arcs are in class W as are all chainable continua [10]. Grispolakis and Tymchatyn have investigated class W from the point of view of hyperspaces [4,5,6]. In [6] they proved that atriodic tree-like continua are in class W, answering a question raised by Ingram in [8]. They also proved in [6] that the Case-Chamberlin continuum (see [1]) is in class W. The Case-Chamberlin continuum is an example of a continuum M, such that every mapping of M to the circle is inessential (a property which is equivalent to M being acyclic in the first Cech cohomology group with integer coefficients) but which admits an essential mapping to the figure eight, and thus the continuum is not tree-like. The Case-Chamberlin continuum is also atriodic.These two theorems of Grispolakis and Tymchatyn thus invite the conjecture that atriodic acyclic continua are in class W. In this paper we show that this is true. We do so without appealing to hyperspace concepts.One might also conjecture that atriodic uni coherent continua are in class W. We exhibit an example to show that this is not true.
Abstract. For the class of hereditarily unicoherent metric continua a spectrum of monotone decompositions has been developed by several authors which " improves" the quotient spaces. This spectrum is developed for a broader class of continua, namely continua with property IUC. A metric continuum M has property IUC provided each proper subcontinuum of M with interior is unicoherent. One important result which develops is that semiaposyndetic IUC continua are hereditarily arcwise connected. Also the notion of smoothness is studied for IUC continua.0. Introduction. In [5] FitzGerald and Swingle, by use of set functions, described a monotone upper semicontinuous decomposition 6Da of a compact Hausdorff continuum M such that sHa is the core decomposition of M with respect to having an aposyndetic quotient space. A spectrum of decompositions which " fills in" between M and M/6i)a has been developed for hereditarily unicoherent metric continua by such authors as J. J. Charatonik [1], E. J. Vought [8, 12] and G. R. Gordh [6,8].The main purpose of this paper is to extend the spectrum of decompositions to the class of IUC continua. A compact metric continuum has property IUC provided each proper subcontinuum with interior is unicoherent [3]. If the continuum has property IUC hereditarily then the continuum has property HIUC. Note that the class of IUC continua includes all atriodic continua and hereditarily unicoherent continua.Throughout this paper M will denote a compact metric continuum. An upper semicontinuous (u.s.c.) decomposition ty of M is core with respect to some property P provided tf) is the unique minimal decomposition with respect to which ty has property P. A set function TV is expansive provided that if A and B are subsets of M then A E N(A) and N(A) E N(B) whenever A E B. The subset A is TV-closed provided A = N(A). In [5, Theorem 2.5, p. 37] FitzGerald and Swingle prove that if TV is any expansive set function on M then there exists a core decomposition § of M with respect to the property: § is u.s.c with TV-closed elements. They also note that if TV is monotone then § is monotone.Let <î> be a u.s.c. decomposition of M and % E 6Í). We denote by %* the subset of M consisting of the sum of the elements of %. If B is a subset of M/ty then B~x will
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.