Janusz R. Prajs

Outlet points and homogeneous continua

I) A proof is presented for Bings conjecture that homogeneous, treelike continua are hereditarily indecomposable. As a consequence, each ho- mogeneous curve admits the continuous decomposition into the maximal termi- nal, homeomorphic, homogeneous, hereditarily indecomposable, treelike sub- continua. (2) A homogeneous, hereditarily unicoherent continuum contains either an arc or arbitrarily small, nondegenerate, indecomposable subcontinua. (3) A treelike continuum with property K which is homogeneous with respect to confluent light mappings contains no two nondegenerate subcontinua with the one-point intersection.

FILAMENT SETS, APOSYNDESIS, AND THE DECOMPOSITION THEOREM OF JONES

Applications of the work introduced by the authors in a recent article, Filament sets and homogeneous continua, are given to aposyndesis and local connectedness. The aposyndetic decomposition theorem of Jones is generalized to spaces with the property of Kelley.

Homogeneous continua in Euclidean

Let X be a homogeneous continuum and let En be Euclidean n-space. We prove that if X is properly contained in a connected (n + 1)manifold, then X contains no n-dimensional umbrella (i.e. a set homeomorphic to the set {(xl,..., Xn+l) e En+: X12 + xn2+l < I and xn+X < 0 and either x = =n = 0 or Xn+1 = 0}) . Combining this fact with an earlier result of the author we conclude that if X lies in En+l and topologically contains En , then X is an n-manifold. The main purpose of this paper is to prove the following theorem. 1. Theorem. Each homogeneous proper subcontinuum of a connected (n + 1)manifold contains no n-dimensional umbrella. The results of this paper are related to two classical results: the first one of S. Mazurkiewicz [M], and, the second one of R. H. Bing [B]. Namely, with the help of the result of [P], we give a full generalization of the result of [B] to all finite-dimensional cases (Theorem 7 below, and also, the statement formulated in the title). As it was emphasized in [P], the theorem of [B] may be obtained by combining two other theorems: 10 each homogeneous locally connected nondegenerate plane continuum is a simple closed curve (this is the result of [M]), 20 each homogeneous plane continuum that contains an arc is locally connected (this is the step really done in [B]), and thus 30 each homogeneous plane continuum that contains an arc is a simple closed curve. (Bings proof did not follow this scheme.) One can easily observe that Theorem 1 implies the result of [M] (for n = 1). Thus this paper generalizes step 10. Step 20 has already been extended in [P] to all finite-dimensional cases. Therefore we get Theorem 7 as a generalization of step 30 . Finally, let us stress the fact that, similarly as in [P], the e-push property (Theorem 4) plays a crucial role in the argument of the proof of Theorem 1. Probably, this is the real reason that the results of [P] and of this paper have not been earlier found. Received by the editors June 9, 1987 and, in revised form, June 1, 1988. 1980 Mathematics Subject Classification (1985 Revision). Primary 54F20; Secondary 54C25.

(n+1)

We prove that each homogeneous continuum which topologically contains an n-dimensional unit cube and lies in (n + l)-dimensional Euclidean space is locally connected. Introduction. Although we still do not have any complete classification of homogeneous plane continua today, there are many significant results concerning homogeneous continua in the planar case. It is interesting to consider which of these results may be naturally extended to higher dimensional cases. In this paper we give an example of such an extension. In 1924 S. Mazurkiewicz [7] proved that a simple closed curve is the only homogeneous plane continuum which is nondegenerate and locally connected. Using this result, R. H. Bing showed in 1959 [1] that a simple closed curve is the only homogeneous plane continuum that contains an arc. All that he had to do to show this (and that he did) was to prove that each homogeneous plane continuum which contains an arc is locally connected. Here we extend the last statement to all finite dimensional cases. MAIN THEOREM. If a homogeneous continuum contains a set homeomorphic to the n-dimensional unit cube and lies in (n + 1)-dimensional Euclidean space, then it is locally connected. It is very natural to ask whether the assumptions of this theorem imply that the considered continuum is an n-manifold. Recently the author has answered this question in the affirmative. This result, which makes heavy use of the Main Theorem, will be presented in a future paper. The general method of the proof of the Main Theorem is similar to that of Bings theorem, i.e. we suppose the existence of a continuum not satisfying this theorem and find a sequence of properties of it. Finally we get two properties such that one contradicts the other. However, only Property 1 here is similar to Property 4 in [1, p. 212], and the particular method of proof here differs much from Bings. There is also another important difference from the Bing method we use many times the so-called s-push property of homogeneous metric continua (see Lemma 4 in [2, p. 37]) which is a corollary to the well-known Effros theorem, and which was not known to Bing in 1959. The author wishes to express profound thanks to Professor J. J. Charatonik for all his help in completing this work. Received by the editors October 15, 1986 and, in revised form, April 10, 1987. 1980 Mathematics Subject Classification (1985 Revision). Primary 54F20; Secondary 54C25.

-space which contain an

Using an example of D. P. Bellamy, we define a 3-composant treelike continuum admitting a fixed-point-free homeomorphism that sends each composant onto itself. This continuum is used to define an indecomposable tree-like continuum that admits a composant-preserving fixed-point-free homeomorphism. A map extension theorem is proved and applied in this construction. Suppose X is a plane continuum and f is a map of X that sends each arccomponent of X into itself. In 1988, Hagopian [3] proved that f has a fixed point if X is tree-like or indecomposable. Solenoids show this is not true for nonplanar indecomposable continua. However, in 1998 Hagopian [4] proved every tree-like continuum has the fixed-point property for arc-component-preserving maps. Recently, Hagopian [5] proved that every composant-preserving map of an indecomposable k-junctioned tree-like continuum has a fixed point. He used tree-chain covers to get his results. Question 1 of [5] asks if this theorem can be generalized to every indecomposable tree-like continuum. We use an example of Bellamy [1] to answer this question in the negative. In fact, we define an indecomposable tree-like continuum that admits a composant-preserving fixed-point-free homeomorphism. It is not known if there exists a tree-like plane continuum or an indecomposable plane continuum that admits a composant-preserving fixed-point-free map. A continuum is a nonempty compact connected metric space. A continuum is indecomposable if it is not the union of two proper subcontinua. Let x be a point of a nondegenerate continuum X. The x-composant of X is the union of all proper subcontinua of X that contain x. If X is indecomposable, then X is the union of uncountably many dense disjoint composants. Given > 0, a mapping f : X → Y is an -mapping if diam(f−1(y)) < for each y ∈ Y . A continuum X is tree-like if for each > 0, there exist a tree Y and an -mapping of X onto Y . A continuum X is tree-like if and only if for each > 0 there is an -tree-chain covering X. A tree-like continuum X is k-junctioned if k is the least integer such that for every positive number there is an -tree-chain covering X with k junction links. We refer to a locally compact noncompact metric space P as a parameter space and denote by P ∪ {∞} the one-point compactification of P . A map α : P → P of a parameter space to itself is called infinity preserving provided that whenever Received by the editors December 8, 2010. 2010 Mathematics Subject Classification. Primary 54F15, 54H25.

n

A continuum K in a space X is said to be semi-terminal if at least one out of every two disjoint continua in X intersecting K is contained in K. Based on this concept, new structural results on Kelley continua are obtained. In particular, two decomposition theorems for Kelley continua are presented. One of these theorems is an improved version of the aposyndetic decomposition theorem for Kelley continua.

-cube are

Known results about lifting of paths for covering, light open and light confluent mappings are in some sense extended for all confluent mappings with the domain being a continuum having the arc property of Kelley. As an application we prove that each confluently tree-like continuum has the fixed point property.