Abstract.Continua admitting arc-structures and arc-smooth continua are introduced as higher dimensional analogues of dendroids and smooth dendroids, respectively. These continua include such spaces as: cones over compacta, convex continua in l2, strongly convex metric continua, injectively metrizable continua, as well as various topological semigroups, partially ordered spaces, and hyperspaces. The arc-smooth continua are shown to coincide with the freely contractible continua and with the metric /T-spaces of Stadtlander. Known characterizations of smoothness in dendroids involving closed partial orders, the set function T, radially convex metrics, continuous selections, and order preserving mappings are extended to the setting of continua with arc-structures. Various consequences of the special contractibility properties of arc-smooth continua are also obtained.Introduction. The purpose of this paper is to introduce and study a well-behaved class of arc-wise connected metric continua called arc-smooth continua) The class of arc-smooth continua, which may be considered as a higher dimensional analogue of the smooth dendroids
Let M \mathfrak {M} denote the class of all hereditarily unicoherent Hausdorff continua in which each indecomposable subcontinuum is irreducible. It is shown that if the continuum M M in M \mathfrak {M} is decomposable, then the set of weak terminal points of M M is a nonempty, proper subset. The following generalization of a theorem of F. Burton Jones is an immediate corollary: if the continuum M M in M \mathfrak {M} is homogeneous, then M M is indecomposable. As an application, it is proved that if X X is a homogenous, hereditarily unicoherent Hausdorff continuum which is an image of an ordered compactum, then X X is an indecomposable metrizable continuum.
Let X denote the limit of an inverse system X -{X a p aa >; A} of locally connected Hausdorff continua. The main purpose of this paper is to define a notion of local connectedness for inverse systems, and to prove that if X is locally connected, then so is the limit X. If the bonding maps p aa > are surjections, then X is locally connected if and only if X is. The following corollaries are obtained. (1) If X is σ-directed and surjective, then X is locally connected. (2) If X is well-ordered, surjective, and weight (X«) ^ λ for each a in A, then either weight (X) ^ λ, or X is locally connected. (3) If X is σ -directed and the factor spaces X α are trees (generalized arcs), then X is a tree (generalized arc). (4) If X is well-ordered and the factor spaces X« are dendrites (arcs), then either X is metrizable, or X is a tree (generalized arc).1. Introduction. By a continuum we mean a compact connected Hausdorff space. Let X denote the limit of an inverse system X = {X α ; p αα A} where the factor spaces X a are locally connected continua, and A is an arbitrary directed set. It is well-known that every continuum X can be obtained as the limit of such a system where the factor spaces are polyhedra (see Theorem 10.1, p. 284, [2]). Hence local connectedness of the factor spaces X a does not imply local connectedness of the limit X. It is the main purpose of this paper to introduce a notion of local connectedness for inverse systems, and to prove that for such systems X the limit space X is locally connected (see Theorem 1). The converse holds if X is a surjective system, i.e., if the bonding maps p aa , are surjections. An immediate corollary is the known result that if X is a monotone inverse system, then X is locally connected [1].In §3 the main theorem is applied to well-ordered and σ-directed inverse systems, i.e., systems in which every countable subset of the index set is bounded above. The following somewhat surprising results are obtained. (1) If the inverse system X is ςr-directed and surjective, then the limit X is locally connected. (2) If X is well-ordered, surjective, and weight (X tt ) = λ for each a in A, then weight (X) Si λ or X is locally connected.Section 4 contains similar results about well-ordered and σ-directed inverse systems of trees (i.e., locally connected, hereditarily unicoherent continua [9]) and generalized arcs (i.e., ordered continua).
Abstract.A branchpoint of a compactum X is a point which is the vertex of a simple triod in X. A surjective map /: X -» Y is said to cover the branchpoints of Y if each branchpoint in Y is the image of some branchpoint in X. If every map in a class % of maps on a class of compacta & covers the branchpoints of its image, then it is said that the branchpoint covering property holds for ff on 0.According to Whyburn's classical theorem on the lifting of dendrites, the branchpoint covering property holds for light open maps on arbitrary compacta. In this paper it is shown that the branchpoint covering property holds for (1) light confluent maps on arbitrary compacta, (2) confluent maps on hereditarily arcwise connected compacta, and (3) weakly confluent maps on hereditarily locally connected continua having closed sets of branchpoints. It follows that the weakly confluent image of a graph is a graph.By a branchpoint of a compactum (i.e., compact metric space) X we mean a point p of X which is the vertex of a simple triod lying in X. Given a surjective map f: X -> Y between compacta we will say that / covers the branchpoints of Y provided each branchpoint of Y is the image under / of a branchpoint of X. Of course, in general, / will not cover the branchpoints of its image, but if it does whenever / is required to be in a particular class ¥ of maps and A" in a particular class G of compacta, then we will say that the branchpoint covering property holds for '$ on G.It is an immediate consequence of Whyburn's classical theorem on the lifting of dendrites under light open maps [8,p.l88] that the branchpoint covering property holds for light open maps on compacta. In this paper we prove that the branchpoint covering property holds more generally for confluent and weakly confluent maps on certain classes of compacta. As a corollary we show that the weakly confluent image of a graph is a graph, thus generalizing a theorem of Whyburn [8,p. 182 confluent.Our first theorem shows that the branchpoint covering property holds for confluent maps on hereditarily arcwise connected compacta (i.e. compacta in which each subcontinuum is arcwise connected).
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