Peano Continua with Unique Symmetric Products

David Herrera-Carrasco, Fernando Macias-Romero, Francisco Vazquez-Juarez
2012 Journal of Mathematics Research
Let X be a metric continuum and n a positive integer. Let F n (X) be the hyperspace of all nonempty subsets of X with at most n points, metrized by the Hausdorff metric. We said that X has unique hyperspace F n (X) provided that, if Y is a continuum and F n (X) is homeomorphic to F n (Y), then X is homeomorphic to Y. In this paper we study Peano continua X that have unique hyperspace F n (X), for each n ≥ 4. Our result generalize all the previous known results on this subject. A continuum is a
more » ... . A continuum is a nondegenerate, compact, connected metric space. A Peano continuum is a locally connected continuum. For a given continuum X and n ∈ N, we consider the following hyperspaces of X F n (X) = {A ⊂ X : A is nonempty and it has at most n points}, and C n (X) = {A ⊂ X : A is closed nonempty and has at most n components}. Both F n (X) and C n (X) are metrized by the Hausdorff metric (Nadler, 1978, Definition 0.1) and are also known as the n-th symmetric product of X and the n-fold hyperspace of X, respectively. When n = 1 it is customary to write C(X) instead of C 1 (X), and refer to C(X) as the hyperspace of subcontinua of X. Let H(X) be any one of the hyperspaces defined above and let K be a class of continua. We say that X ∈ K has unique hyperspace H(X) in K if whenever Y ∈ K is such that H(X) is homeomorphic to H(Y), it follows that X is homeomorphic to Y. If K is the class of all continua, we simply say that X has unique hyperspace H(X). The topic of this paper is inserted in the following general problem. Problem. Find conditions, on the continuum Z, in order that Z has unique hyperspace H(Z). A finite graph is a continuum that can be written as the union of finitely many arcs, each two of which are either disjoint or intersect only in one or both of their end points. Let It has been proved the following results (a)-(m). (a) If X ∈ G different from an arc or a simple closed curve, then X has unique hyperspace C(X), see Duda (1968, p. 265-286) and Acosta (2002, p. 33-49). (b) If X ∈ G, then X has unique hyperspace C 2 (X), see Illanes (2002(2) , p. 347-363). (c) If X ∈ G, then X has unique hyperspace C n (X) for each n ∈ N − {1, 2}, see Illanes (2003, p. 179-188). (d) If X ∈ G, n, m ∈ N, Y is a continuum and C n (X) is homeomorphic to C m (Y), then X is homeomorphic to Y, see Illanes (2003, p. 179-188). 1