Quotients of Bing Spaces

A Bing space is a compact Hausdorff space whose every component is a hereditarily indecomposable continuum. We investigate spaces which are quotients of a Bing space by means of a map which is injective on components. We show that the class of such spaces does not include every compact space, but does properly include the class of compact metric spaces. Our entire development is based on Krasinkiewicz’s penetrating notions of folding and double pairs. In section 1, we provide the definitions of double pairs and folding and show how these concepts are related to hereditarily indecomposable continua. Section 2 is the heart of this paper. We begin with a fixed compact space X. In Proposition 2.4, a simple two dimensional construction of a subspace Y of X × [−1, 1] and the properties of Y are presented. In Lemma 2.8 the components of Y are characterized in terms of components of X and certain subspaces of X. In section 3, we use the machinery developed in section 2 to describe when a selection of components from Y , one for each component in X, can be made in such a way that their union Z is compact (see Theorem 3.5 and Corollary 3.7). This leads rather quickly to Theorem 3.17 and Corollary 3.18, which give classes of spaces (including compact metric spaces and connected spaces) which are guaranteed to be quotients of a Bing space by means of a map which is injective on components. We conclude by showing (Theorem 3.21) that there are spaces which are not such quotients of the above sort. Throughout this paper, all topological spaces will be assumed to be Hausdorff without further mention. Since our development focuses on compact Hausdorff spaces, spaces not otherwise specified will be assumed to be compact. 1. Double pairs, folding and continua We begin with a review of some of the basic notions. This section owes a particularly heavy debt to Krasinkiewicz [3] where the notion of a “double pair” is fundamental. The pairs used in the current paper are composed of open sets (rather than closed sets as in [3]), but we do not change the terminology and continue to call them double pairs. Throughout this section, X and Y denote compact spaces. Definition 1.1. A double pair is an ordered pair ((A−1, B−1) , (A1, B1)) of ordered pairs (Ai, Bi) of open subsets of X such that Ai ⊆ Bi and Ai ∩ Bj = ∅ for i, j ∈ {−1, 1}, i 6= j. Throughout this paper, we adopt the convention that i ∈ {−1, 1} and j = −i. For technical reasons (see Lemma 3.2 below) we allow the Ai’s and Bi’s to be empty. Of course, A denotes the closure of A. Date: August 25, 2006. 2000 Mathematics Subject Classification. Primary 54F15 ; Secondary 54D15,54B15.