Continuous images of arcs

Abstract

ABSTRACT. A characterization of continuous images of (Hausdorff) arcs is used to describe their cyclic elements as inverse limits of inverse systems of 'nice' spaces with 'nice' bonding mappings. Introduction.By a continuum we will mean a compact and connected Hausdorff space, and an arc will be understood as a Hausdorff arc, i.e., a continuum with exactly two non-cut-points.Recall that arcs are precisely compact and connected linearly ordered topological spaces, and each separable arc is homeomorphic to [0,1] (see for example [6]).The classical Hahn-Mazurkiewicz theorem [5,10] characterizes continuous images of [0,1] (in the class of all Hausdorff spaces) as locally connected and metrizable continua. Since 1960 it is known that there exist locally connected continua which are continuous images of no arc ([7]; recall that each Hausdorff space which is a continuous image of some arc is a locally connected continuum-see for example [6]). Hence there is no straightforward generalization of the Hahn-Mazurkiewicz theorem. The first reasonable characterization of continuous images of arcs was found in 1985 in [12], where several natural equivalent conditions were proved. Some of those conditions involve the use of the cyclic element theory (see for example [20] for the cyclic element theory in metrizable locally connected continua, and [21,2] for generalizations of the theory to the class of all locally connected continua). The utility of the cyclic element theory as a way to find a nonseparable version of the Hahn-Mazurkiewicz theorem was noticed for the first time in [2], where it was proved that if each cyclic element of a locally connected continuum A is a continuous image of some arc, then also the whole A is an image of some arc.In the present paper we apply the characterization theorem of [12] to prove that if A is a cyclic element of a continuum which is a continuous image of some arc, then X is homeomorphic to the inverse limit of an inverse sequence of some 'piecewise' metrizable continua (each of which is also a continuous image of some arc and, moreover, can be defined by a simple induction) with natural bonding mappings. Moreover, we describe some examples and applications of the obtained results. Some further applications of the related methods can be found in [14].