The Freudenthal compactification of tree-like generalized continua

Tree-likeness of generalized continua is defined by means of inverse limits of locally finite trees with proper bonding maps. The main theorem of this paper shows that the Freudenthal compactification preserves and reflects tree-likeness. Some consequences of interest are given.Ministerio de Ciencia e Innovació

Metrics defined via discrepancy functions

Abstract We introduce the notion of a discrepancy function, as an extended real-valued function that assigns to a pair (A, U ) of sets a nonnegative extended real number ω(A, U ), satisfying specific properties. The pairs (A, U ) are certain pairs of sets such that A ⊆ U , and for fixed A, the function ω takes on arbitrarily small nonnegative values as U varies. We present natural examples of discrepancy functions and show how they can be used to define traditional pseudo-metrics, quasimetrics and metrics on hyperspaces of topological spaces and measure spaces

Topology Proceedings

ABSTRACT. We show that for any continua X and Y the smoothness of either the hyperspace C(X) or 2 x or of the Cartesian product X x Y implies the ptoperty of Kelley for X. An example is constructed showing that the converse is not true. A continuum is a compact connected metric space. Given a point p E X and a positive number r we denote by Bx(p, r) the open ball with center p and radius r and, for A c X we defi~e Nx(A,r) == U{B(x,r): x E A}. We say that continuum X has the property of J<elley if for each £ > 0 there is a 8 > 0 such that for each point x EX, for 1991 Mathematics Subject Classification. 54B10, 54B20, 54F15. !(ey 1vords and phrases. continuum, hyperspace, product, property of Kelley, sll10oth

OPENNESS AND MONOTONEITY OF INDUCED MAPPINGS

Abstract. It is shown that for locally connected continuum X if the induced mapping C(f):C(X)→C(Y)isopen,thenfis monotone. As a corollary it follows that if the continuum X is hereditarily locally connected and C(f) is open, then f is a homeomorphism. An example is given to show that local connectedness is essential in the result. All spaces considered in this paper are assumed to be metric. A mapping means a continuous function. We denote by N the set of all positive integers, and by C the complex plane. Given a space S, a point c ∈ S and a number ε>0, we denote by BS(c, ε) the open ball in S with center c and radius ε. A continuum means a compact connected space. Given a continuum X with ametricd,welet2Xdenote the hyperspace of all nonempty closed subsets of X equipped with the Hausdorff metric H defined by H(A, B) =max{sup{d(a, B):a∈A},sup{d(b, A):b∈B}} (see, e.g., [5, (0.1), p. 1 and (0.12), p. 10]). Further, we denote by C(X) the hyperspace of all subcontinua of X, i.e., of all connected elements of 2X,andby F1(X) the hyperspace of singletons. The reader is referred to Nadler’s book [5] for needed information on the structure of hyperspaces. Given a mapping f: X → Y between continua X and Y, we consider mappings (called the induced ones) defined by 2 f:2 X →2 Y 2 f (A) =f(A) for every A ∈ 2