The hyperspace of connected boundary subcontinua of a continuum

doi:10.1016/j.topol.2020.107573

Topology and its Applications

Volume 290, 1 March 2021, 107573

https://doi.org/10.1016/j.topol.2020.107573 Get rights and content

Abstract

Given a metric continumm X, let $C B (X)$ be the hyperspace of subcontinua of X with connected boundary. In this paper we present results concerning $C B (X)$ , first about continua for which every subcontinuum has connected boundary. Then we study continua that only have one-point sets as connected boundary proper subcontinua. Later we include characterizations of arcs and simple closed curves. Finally we give a formula for the number of components of $C B (X)$ for finite graphs.

Introduction

A continuum is a nonempty, compact, connected metric space. If X is a continuum, we denote the hyperspace of all subcontinua of X by $C (X)$ , the hyperspace of all meager subcontinua of X by $M (X)$ , and the hyperspace of all connected boundary subcontinua of X by $C B (X)$ ; that is, $C (X) = {A \subset X : A is nonempty, closed and connected}$ , $M (X) = {A \in C (X) : b d (A) = A}$ and $C B (X) = {A \in C (X) : b d (A) is connected}$ . We denote by $F_{1} (X)$ the hyperspace of singletons of X. These hyperspaces are considered with the Hausdorff metric, [8, p. 1]. Notice that $F_{1} (X) \subset M (X) \subset C B (X) \subset C (X)$ and $X \in C B (X) - M (X)$ . In [10] the author presented a study of the hyperspace $M (X)$ . In [3] P. Krupski study the Borel complexity of the hyperspace of proper subcontinua of X with connected boundary, mainly for π-Euclidean Peano continua. In this paper we present general properties of the hyperspace $C B (X)$ . In Section 3 we study continua for which every subcontinuum has connected boundary, that is, continua X having the property that the hyperspaces $C B (X)$ and $C (X)$ coincide. In Section 4 we characterize continua having only one-point sets as connected boundary proper subcontinua, that is continua X such that the hyperspace $C B (X)$ is equal to $F_{1} (X) \cup {X}$ . In Section 5, we present characterizations of the arc and the simple closed curve by using the hyperspace of connected boundary subcontinua. In Section 6 we give a formula for the number of components of the hyperspace $C B (X)$ when X is a finite graph.

Section snippets

Definitions

A set is said to be nondegenerate if it consists of more than one point. We say that A is a proper subset of X provided that A is contained in X and A is different from X.

A continuum X is unicoherent provided that whenever A and B are closed connected subsets of X such that $X = A \cup B$ , then $A \cap B$ is connected; X is hereditarily unicoherent if each subcontinuum of X is unicoherent.

A continuum is decomposable provided that it can be written as the union of two of its proper subcontinua; otherwise it is

Continua having only connected boundary subcontinua

In this section we study properties of continua having only connected boundary subcontinua; that is continua X for which the equality $C B (X) = C (X)$ holds.

Theorem 3.1

For a continuum X the following conditions are equivalent:

(1)
Every subcontinuum of X has connected boundary;
(2)
X is unicoherent and, for every subcontinuum A of X, $\overline{X - A}$ is connected.

Proof

$(1) \to (2)$ Suppose X is not unicoherent. Let A and B be subcontinua of X such that $X = A \cup B$ and $A \cap B$ is not connected. Put $A \cap B = H \cup K$ , where H and K are nonempty closed and disjoint

Continua having only one-point sets as connected boundary proper subcontinua

In this section we present conditions equivalent to having only one-point sets as connected boundary proper subcontinua, we mean conditions for a continuum X in order to satisfy $C B (X) - {X} = F_{1} (X)$ .

Theorem 4.1

For a continuum X the following conditions are equivalent:

(1)
Each connected boundary proper subcontinuum of X is a one-point set;
(2)
X is hereditarily locally connected and does not contain meager arcs nor cut points.
(3)
Each meager subcontinuum of X is a one-point set and X does not contain cut points.

Proof

$(1) \to (2)$ If X

Characterizations of arcs and simple closed curves

Proposition 5.1

If X is a continuum whose hyperspace of meager subcontinua is a one-dimensional space, then X is hereditarily locally connected.

Proof

If X is not hereditarily locally connected, then X contains a convergence continuum A [9, Theorem 10.4]. So, $C (A)$ is a subset of $M (X)$ . It is known that the dimension of $C (A)$ is at least 2, [1, Theorem 1]. It follows that $M (X)$ is not a one-dimensional space. □

Since, for any continuum X, the hyperspace $M (X)$ is a subset of $C B (X)$ we have the following corollary.

Corollary 5.2

If X is a

Counting components of $C B (X)$

Throughout this section X will denote a finite graph. Given a positive integer n, a simple n-od is a finite graph which is the union of n arcs emanating from a single point, v, and otherwise disjoint from one another. The point v is called the core of the simple n-od. We note that a 3-od is a simple triod. We say that a point p of X is of order n in X, denoted by ${ord}_{X} (p) = n$ , if p has a closed neighborhood which is homeomorphic to a simple n-od having p as the core. If ${ord}_{X} (p) = 1$ , then we say that

Acknowledgements

The authors wish to thank the participants at the Workshops on Continuum Theory and Hyperspaces, organized by Alejandro Illanes and Verónica Martínez de la Vega in México, in 2014, 2016 and 2017, for useful discussions on the topic of this paper, in particular to Pawel Krupski for the discussions in the workshop in 2016.

References (12)

A. Illanes et al.
Weak n-ods, n-ods and strong n-ods
Topol. Appl.
(2017)
P. Krupski
Hyperspaces of continua with connected boundaries in π-Euclidean Peano continua
Topol. Appl.
(2020)
V. Martínez-de-la Vega et al.
Embedding hyperspaces
Topol. Appl.
(2012)
C. Eberhart et al.
The dimension of certain hyperspaces
Bull. Pol. Acad. Sci.
(1971)
K. Kuratowski
Topology
(1968)
T. Maćkowiak
Continuous mapping on continua
Diss. Math.
(1979)

There are more references available in the full text version of this article.

Cited by (0)

View full text

The hyperspace of connected boundary subcontinua of a continuum

Abstract

Introduction

Section snippets

Definitions

Continua having only connected boundary subcontinua

Continua having only one-point sets as connected boundary proper subcontinua

Characterizations of arcs and simple closed curves

Counting components of CB(X)

Acknowledgements

Topol. Appl.

Topol. Appl.

Topol. Appl.

The dimension of certain hyperspaces

Bull. Pol. Acad. Sci.

Topology

Continuous mapping on continua

Diss. Math.

Counting components of $C B (X)$