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
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).
In this paper, certain results of Bing (1) and myself (2) are extended. It is well-known that a chainable compact metric continuum must be a-triodic (contain no triods), hereditarily unicoherent (the common part of each two subcontinua is connected), and each subcontinuum must be chainable. Our principal result states that a compact metric continuum M is chainable if and only if M is a-triodic, hereditarily unicoherent and each indecomposable subcontinuum of M is chainable. Some condition on the indecomposable subcontinua of M seems essential, if we consider the dyadic solenoid, 5, which is indecomposable, a-triodic and hereditarily unicoherent. Indeed, each proper subcontinuum of S is an arc. However, S is not chainable, since it cannot be embedded in the plane.
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