An order theoretic characterization of nested spaces
Partial orders in topological spaces have been of great interest to topologists since the classic work of Nachbin [7]. Later L. E. Ward, Jr. [11] studied partially ordered topological spaces and in 1954 gave a characterization of trees in terms of an inherent partial order [12]. Recently Fugate, Gordh and Lum [1] and [2] gave an order theoretic characterization of arc smooth continua. Other similar results were obtained by Smithson [8], Smithson and Muenzenberger [4], and Ward [13]. Another area in which partial orders have played a key role is the study of dendritic spaces. See, in particular, the comprehensive work of Ward [14] and the recent results of Muenzenberger and Smithson [5] and [6]. In 1946, G. S. Young, Jr. [15] introduced the notion of a nested space. Nested spaces possess a natural order structure which has proven useful in obtaining fixed point theorems (see, for example, [9] and [10]). The algebraic properties of nested spaces were studied extensively by Muenzenberger and Smithson [3]. The purpose of the present paper is to give a characterization of nested spaces analogous to Ward's characterization of trees [12]. Suppose (X, 6) is a space with a partial order < . We set L(x) = {y: y < x} and M(x) = {y: x < y}. A set A c X is an antiset in case no two elements of A are related. Further, the partial order is order dense in case x <y implies that there isaz E Xwithx <z <y. In this paper a continuum is a compact, connected T2-space and an arc is a continuum which contains exactly two nonseparating points. A tree is a continuum in which any two distinct points can be separated by the omission of a third point. Then a space X is nested in case it is arcwise connected, T2 and the union of every nest of arcs is contained in an arc. If X is a nested space and a, b E X, then there is exactly one arc in X with endpoints a, b. Using this fact we define a partial order < on X as follows. Let e E X be fixed. Then x < y iff x is an element of the arc with endpoints e, y. A number of properties of this partial order can be derived and
Arc-Smooth Continua
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 12, 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 K-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 [8], includes (a) smooth dendroids; (b) convex (more generally star-like) continua in 12; (c) cones over compacta; (d) strongly convex metric continua; (e) injectively metrizable continua [17]; (f) continua ruled by arcs [22]; (g) continuumfi semilattices with unit; (h) various hyperspaces of continua; and (i) certain partially ordered spaces. The arc-smooth continua coincide with the (a) freely contractible continua [17]; (b) metric K-spaces [36]; and (c) ruled metric spaces [9]. Our study of these continua involves a new concept called an arc-structure which is patterned after the "natural arc-structure" possessed by dendroids. In order to motivate the definition, we begin by considering the special case of dendroids. Let X be a dendroid (i.e., an arc-wise connected, hereditarily unicoherent metric continuum). Each pair of points x =#y in X determines a unique arc from x to y denoted by xy. By convention, xx = {x}. Denote by C(X) the hyperspace of subcontinua of X with the Hausdorff metric (see [19]). The function A: X x X -> C(X) defined by A (x, y) = xy satisfies the following metric-like axioms for all x, y and z in X.
A fixed point theorem for plane continua
In this paper it is proved that every bounded arcwise connected plane continuum which does not separate the plane has the fixed point property. A set X is said to have the fixed point property if each continuous function ƒ on X into itself leaves some point fixed (that is, there is a point x belonging to X such that f(x)=x). The problem "Must a bounded plane continuum which does not separate the plane have the fixed point property?" has motivated a great deal of research in plane topology. K. Borsuk in 1932 proved that every Peano continuum which lies in the plane and does not separate the plane has the fixed point property [2]. Since that time, other general conditions have been found which insure that a plane continuum has this property. In 1967, H. Bell proved that every bounded plane continuum which does not separate the plane and has a hereditarily decomposable boundary has the fixed point property [ l ] . The following question is still outstanding. If a bounded plane continuum is arcwise connected and does not separate the plane, then must it have the fixed point property? Here an affirmative answer is given to this question by proving the following theorem. If M is a bounded arcwise connected plane continuum which does not have infinitely many complementary domains, then the boundary of M does not contain an indecomposable continuum. Throughout this paper 5 is the set of points of a simple closed surface (that is, a 2-sphere). The proof of the following theorem is based on techniques which are closely related to the folded complementary domain concept defined by F. Burton Jones [3, p. 173]. THEOREM 1. Suppose M is a continuum in Sy S—M does not have AMS 1970 subject classifications. Primary 54H25, 57A05, 47H10, 55C20; Secondary 54F22.