Hereditarily irreducible maps

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Abstract

A map f:XY from a continuum X onto a continuum Y is said to be hereditarily irreducible, if f(A)f(B) for any subcontinua A and B such that AB. We investigate properties of hereditarily irreducible maps between continua. Special attention is given to maps between graphs and maps from the interval.

Introduction

The notion of hereditarily irreducible map generalizes the notion of an arcwise increasing maps. In fact the two notions coincide if the domain is an arc. Therefore our investigation of hereditarily irreducible map is an extension of the work done by B. Espinoza and E. Matsuhashi [4]. We generalize some of their theorems to more general settings and we answer a problem from their article.

In Section 3, we introduce a new notion of an order of a point and compare it to the order in the classical sense. The two notions agree on graphs, but they differ even for locally connected continua. Then we investigate hereditarily irreducible maps between graphs. Section 4 is devoted to the existence of hereditarily irreducible maps from the arc. Such maps were introduced in [5] under the name of “arcwise increasing maps”; further investigation was done in [4]. We find necessary and sufficient conditions for existence of such maps. In Section 5 and 6, we investigate hereditarily irreducible maps between graphs and onto dendrites, respectively. We devote Section 7 to open and hereditarily irreducible maps. We show that such maps must be homeomorphisms if the domain is a locally connected continuum, but they do not need to be homeomorphisms if we relax the assumption of locally connectedness of the domain. Finally, in Section 8, we give an answer to a question by Espinoza and Matsuhashi in [4].

Section snippets

Preliminaries

In this section, we introduce notions that will be used throughout this paper. A space X is called a continuum if it is non-empty compact connected metric space. A subset of a continuum X which is itself a continuum is called a subcontinuum of X. A continuum is non-degenerate if it contains more than one point. An arc is a homeomorphic image of the closed unit interval [0,1] and a simple closed curve is a homeomorphic image of a circle.

If X is a continuum and h:[0,1]X is a homeomorphism onto

Order of a point and functions between graphs

In this section, we introduce a new concept of an order of a point. It generalizes the concept of order in the classical sense. The two concepts coincide for graphs, but are different even for locally connected continua.

Let us recall the classical definition of an order.

Definition 3.1

Let X be a continuum and p be a point in X, and let α be a cardinal number. We say that ordcX(p)α if there are arcs Aγ for 0γα in X such that p is an endpoint of Aγ and AγAδ={p} for 0γ<δα. Finally, we let ordcX(p)=α if ordc

Hereditarily irreducible maps from an arc

In this section, we give two characterizations of hereditarily irreducible images of an arc. In the case of the image is a graph we have a full characterization by Theorem 3.20. Also if the image is a continuum that does not contain free arcs, we always have such mapping by [4, Theorem 4.21, p. 87].

Theorem 4.1

Suppose X and Y are locally connected continua such that YX. If Y contains all free arcs of X and there is a hereditarily irreducible map from [0,1] onto Y, then there is a hereditarily irreducible

Hereditarily irreducible images of graphs

In [4] the authors proved that for every locally connected continuum X without free arcs there is a hereditarily irreducible map from [0,1] onto X. The goal of this section is to generalize the result to have any graph in the domain, not just [0,1].

We have to start with the definitions of necessary symbols. Let X and Y be continua, x¯={xi}i=1nX, y¯={yi}i=1nY be any subsets with n elements, not necessarily different, and let C(X,Y) denote the set of all maps from X to Y. We define the

Hereditarily irreducible maps onto dendrites

Theorem 6.1

Let D be a dendrite containing no copy of D3. If there is a hereditarily irreducible map f from a locally connected continuum X onto D, then f is a homeomorphism.

Proof

Let x1,x2 be two distinct elements of X, and let A be an arc in X irreducible between {x1,x2}. Since D is a dendrite containing no copy of D3, f(A) contains no simple closed curve and no copy of D3. By Proposition 4.7, f is a homeomorphism on A. So, f(x1)f(x2), and therefore, f is a one-to-one on X. Since f is one-to-one map from a

Mappings

Theorem 7.1

Let f be a hereditarily irreducible map from a locally connected continuum X onto a locally connected continuum Y, then f is an open map if only if f is a homeomorphism.

Proof

Suppose, on the contrary, that f is not a homeomorphism. Then by our assumption, there are x1 and x2 in X such that x1x2 and f(x1)=f(x2)=y. Let Z be a continuum neighborhood of y in Y such that x1 and x2 belong to different components of f1(Z). Let C1 and C2 be components of f1(Z) that contain x1 and x2 respectively. Since f

Answers

In [4, Theorem 4.21.3, Question 6.4, p. 92] B. Espinoza and E. Matsuhashi proved that for any locally connected continuum X contains no free arcs and for any two points p,qX, there is a hereditarily irreducible map f from [0,1] onto X such that f(0)=p and f(1)=q and posed a question: Is the converse of this Theorem true? We answered their question in the positive.

Theorem 8.1

If X is a locally connected continuum and for any two points p,qX there is a hereditarily irreducible map f from [0,1] onto X such

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