A partition theorem for collections of universal subcontinua

Abstract

This paper is concerned with conditions under which a collection of universal subcontinua may be partitioned into finitely many subcollections, each of which has the finite intersection property. Throughout X will denote a Hausdorff topological space. A continuum A c X will be called a universal subcontinuum of (USC) X if A N B is a continuum for each continuum B c X. (Ordinarily it is assumed that X is a continuum, but this assumption is unnecessary for our purposes.) WALLACE, [3], studied USC under the title semi-chains. We are concerned with the following topological-combinatorial problem. If ~ is a collection of universal subcontinua, if n_->2 is a positive integer, and if of each n distinct elements of ct at least two have a non-empty intersection, does there exist an integer m <_- n -- 1 and m subcollections cq ..... at,, of ~, each of which has the finite intersection property, such that ct = ~ U -.- U ~t,,? We have not been able to answer this question in general, but in this paper we will answer it affirmatively for n = 3. The importance of the present problem lies in the fact that if ct is a collection of USC which has the finite intersection property then the intersection over ~ is a non-empty USC. The theorem presented here is of the type which has traditionally played a role in the study of the action of certain types of mappings on Hausdorff continua. It applies, for example, to collections of (i) compact nodal sets in a connected space, (ii) subcontinua of a hereditarily unicoherent continuum, and (iii) maximal cyclic elements in a peano continuum, [4]. It also applies to compact segments of the real line, where it overlaps a theorem of DEBRUNNER and HADWIGER [1] ; however, this special case can be treated by simpler methods than those used here. The following lemma is essentially lemma 1 of [2]. However, the proof is given here for the sake of completeness.