Hereditarily irreducible maps
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 and a simple closed curve is a homeomorphic image of a circle.
If X is a continuum and 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 if there are arcs for in X such that p is an endpoint of and for . Finally, we let if
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 . If Y contains all free arcs of X and there is a hereditarily irreducible map from 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 onto X. The goal of this section is to generalize the result to have any graph in the domain, not just .
We have to start with the definitions of necessary symbols. Let X and Y be continua, , be any subsets with n elements, not necessarily different, and let 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 . If there is a hereditarily irreducible map f from a locally connected continuum X onto D, then f is a homeomorphism.
Proof Let be two distinct elements of X, and let A be an arc in X irreducible between . Since D is a dendrite containing no copy of , contains no simple closed curve and no copy of . By Proposition 4.7, f is a homeomorphism on A. So, , 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 and in X such that and . Let Z be a continuum neighborhood of y in Y such that and belong to different components of . Let and be components of that contain and 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 , there is a hereditarily irreducible map f from onto X such that and 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 there is a hereditarily irreducible map f from onto X such
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2023, Memoirs of the American Mathematical Society