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Based on the past study of homogeneous continua, this paper concludes that compact connected metric topological groups and isometrically homogeneous continua fall into the following three mutually disjoint classes: (1) indecomposable; (2) aposyndetic and semi-indecomposable; (3) mutually aposyndetic. Among all continua these are special classes with members having extremal properties. Indecomposable isometrically homogeneous continua are characterized as solenoids, and one-dimensional isometrically homogeneous continua are characterized as solenoids or circles. It is shown that path connected isometrically homogeneous continua are locally connected.

The main result of this paper states that every homogeneous pseudo-path connected continuum is weakly chainable, or equivalently, every homogeneous continuum connected by continuous images of the pseudo-arc is itself a continuous image of the pseudo-arc. We notice that even though there exist homogeneous path connected continua that are not continuous images of an arc (Prajs, 2002), they all are continuous images of the pseudo-arc.

We establish equivalences between compacta that admit mappings that limit to the identity, and compacta that are inverse limits of the images under these maps. Our results have relationships to Mardesic and Segalʼs equivalence between polyhedra-like compacta and inverse limits of polyhedra, to the Anderson–Choquet Embedding Theorem, to approximative absolute neighborhood retracts, and to continua that are approximated from within as defined by C.A. Eberhart and J.B. Fugate.

A continuum K K in a space X X is said to be semi-terminal if at least one out of every two disjoint continua in X X intersecting K K is contained in K 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.

A new decomposition, the mutually aposyndetic decomposition of homogeneous continua into closed, homogeneous sets is introduced. This decomposition is respected by homeomorphisms and topologically unique. Its quotient is a mutually aposyndetic homogeneous continuum, and in all known examples, as well as in some general cases, the members of the decomposition are semiindecomposable continua. As applications, we show that hereditarily decomposable homogeneous continua and path connected homogeneous continua are mutually aposyndetic. A class of new examples of homogeneous continua is defined. The mutually aposyndetic decomposition of each of these continua is non-trivial and different from Jones’ aposyndetic decomposition.

We investigate the structure of the collection of terminal subcontinua in homogeneous continua. The main result is a reduction of this structure to six specific types. Three of these types are of one-dimensional spaces, and examples representing these types are known. It is not known whether higher dimensional examples having non-trivial terminal subcontinua and representing the three remaining types exist.

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.

This paper applies the concepts introduced in the article: Filament sets and homogeneous continua [J.R. Prajs, K. Whittington, Filament sets and homogeneous continua, Topology Appl. 154 (8) (2007) 1581–1591, doi:10.1016/j.topol.2006.12.005 ] to decompositions of homogeneous continua. Several new or strengthened results on aposyndesis are given. Newly defined decompositions are discussed. A proposed classification scheme for homogeneous continua is shown to be mostly invariant under Jones' aposyndetic decomposition.

This paper is a classification study of homogeneous continua. The focus is on homogeneous continua that are filament additive, i.e., those whose every two filament subcontinua have filament union. We show that a homogeneous filament additive continuum that is either path connected or hereditarily decomposable is locally connected. This allows us to partially answer two classic questions about homogeneous continua. We also prove several natural conditions equivalent to the filament additivity of a homogeneous continuum. An outline is given of a classification scheme for homogeneous continua.

We establish equivalences between compacta that admit a sequence of retractions that converge uniformly to the identity map and compacta that are inverse limits on subcompacta with retractions for bonding maps. We give partial answers to questions of Charatonik and Prajs, and of Krasinkiewicz. Our results are related to and use results from another paper of the authors \cite{mp}.

Continua that are approximative absolute neighborhood retracts (AANR's) are characterized as absolute terminal retracts, i.e., retracts of continua in which they are embedded as terminal subcontinua. This implies that any AANR continuum has a dense arc component, and that any ANR continuum is an absolute terminal retract. It is proved that each absolute retract for any of the classes of: tree-like continua, λ-dendroids, dendroids, arc-like continua and arc-like λ-dendroids is an approximative absolute retract (so it is an AANR). Consequently, all these continua have the fixed point property, which is a new result for absolute retracts for tree-like continua. Related questions are asked.

We define an unusual continuum M with the fixed-point property in the plane R. There is a disk D in R such that M ∩ D is an arc and M ∪ D does not have the fixed-point property. This example answers a question of R. H. Bing. The continuum M is a countable union of arcs.

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.

For any class K of compacta and any compactum X we say that: (a) X is conuently K-representable if X is homeomorphic to the inverse limit of an inverse sequence of members of K with conuen t bonding mappings, and (b) X is conuently K-like provided that X admits, for every " > 0, a conuen t "-mapping onto a member ofK. The symbolLC stands for the class of all locally connected compacta. It is proved in this paper that for each compactum X and each familyK of graphs, X is conuen tlyK- representable if and only if X is conuen tlyK-like. We also show that for any compactum the properties of: (1) being conuen tly graph-representable, and (2) being 1-dimensional and conuen tly LC-like, are equivalent. Consequently, all locally connected curves are conuen tly graph-representable. We also conclude that all conuen tly arc-like continua are homeomorphic to inverse limits of arcs with open bonding mappings, and all conuen tly tree-like continua are absolute retracts for hereditarily unicoherent continua.

We investigate absolute retracts for hereditarily unicoherent continua, and also the continua that have the arc property of Kelley (i.e., the continua that satisfy both the property of Kelley and the arc approximation property). Among other results we prove that each absolute retract for hereditarily unicoherent continua (for tree-like continua, for -dendroids, for dendroids) has the arc property of Kelley.

Given a homogeneous compactum X , a general method of constructing homogeneous compacta as inverse limit spaces lim{ X n , b n } is introduced, where X n are covering spaces for X and b n are covering mappings. If [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /] is the Menger universal curve, a variant of this construction provides a homogeneous, arcwise connected, non-locally connected curve. This answers a question of K. Kuperberg.

A collection of continua is described such that each continuum either is incomparable with some element of the collection, or it is a continuous image of the harmonic fan. As a consequence, a characterization of continua comparable with all continua is obtained.

We provide a structural characterization of all continua X whose hyperspace C ( X ) of all subcontinua is the countable union of Peano continua. Applying this result we prove that there exists a uniformly path connected continuum X with no continuous mapping from C ( X ) onto X.

We prove an early announcement by Knaster on a decomposition of the plane. Then we establish an announcement by Anderson saying that the plane annulus admits a continlous decomposition into pseudo-arcs such that the quotient space is a simple closed curve. This provides a new plane curve, "a selectible circle of pseudo-aics", and answers some questions of Lewis.

It is proved that a complete metric space topologically contains the Sierpiński universal plane curve if and only if it has a subset with so-called bypass property, i.e. it has a subset K K containing an arc such that for each a ∈ K a\in K and for each open arc A ⊂ K A\subset K with a ∈ A a\in A , there exists an arbitrary small arc in K ∖ { a } K\setminus \{a\} joining the two components of A ∖ { a } A\setminus \{a\} .

Among other things it is proved that the set of all open mappings between compacta X and Y is topologically complete if X is locally connected and Y is a graph, and this set is not topologically complete if it is nonempty and Y is a manifold of dimension >1, or Y is the Menger universal curve, or Y is a pseudo-arc.

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

Publisher Summary This chapter discusses 30 open problems in the theory of homogeneous continua. The problems collected are divided into two parts. In Section 2 miscellaneous problems are presented, some of which already have been published, and some are new. In the author's view, these questions may have potential to become a part of the mainstream study of homogeneous continua in the future. The remaining part of the paper is devoted to a new line of study of homogeneous continua, based on the duality of filament and ample subcontinua. This new research is related to the past applications of aposyndesis to homogeneous continua. The idea is to identify major archetypical classes of homogeneous continua related to the structure of their filament subcontinua, and investigate properties of these classes.

We investigate absolute retracts for classes of hereditarily unicoherent continua, tree-like continua, λdendroids, dendroids and some other related ones. The main results are: (1) the inverse limits of trees with confluent bonding mappings are absolute retracts of hereditarily unicoherent continua; (2) each tree-like continuum is embeddable in a special way in a tree-like absolute retract for the class of hereditarily unicoherent continua; (3) a dendroid is an absolute retract for hereditarily unicoherent continua if and only if it can be embedded as a retract into the Mohler-Nikiel universal smooth dendroid.

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 Bing's 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 Bing's. 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.

Hagopian has characterized solenoids as nondegenerate homogeneous continua such that each of their proper nondegenerate subcontinua is an arc. In this paper it is shown that solenoids are exactly these nondegenerate continua with arcs as all (sufficiently small) proper nondegenerate subcontinua, which are homogeneous with respect to open (light open) mappings.

Let X X be a continuously homogeneous Hausdorff continuum. We prove that if there is a sequence A 1 , A 2 , … {A_1},{A_2}, \ldots of its arc components with X = c1 A 1 ∪ c1 A 2 ∪ ⋯ X = {\text {c1}}{A_1} \cup {\text {c1}}{A_2} \cup \cdots , and there is an arc component of X X with nonempty interior, then X X is arcwise connected. As consequences and applications we get: (1) if X X is the countable union of arcwise connected continua, then X X is arcwise connected; (2) if X X is nondegenerate and metric, the number of its arc components is countable and it contains no simple triod, then it is either an arc or a simple closed curve; and, in particular, (3) an arc is the only nondegenerate continuously homogeneous arc-like metric continuum with countably many arc components.

New tools are introduced for the study of homogeneous continua. The subcontinua of a given continuum are classified into three types: filament , non-filament , and ample , with ample being a subcategory of non-filament. The richness of the collection of ample subcontinua of a homogeneous continuum reflects where the space lies in the gradation from being locally connected at one extreme to indecomposable at another. Applications are given to the general theory of homogeneous continua and their hyperspaces.