On monotone unions of manifolds

Abstract

If U is a connected separable metric space that is locally euclidean of dimension n, U is an n-manifold. We do distinguish n-manifolds and n-manifolds with boundary. By 1(U) we denote the collection of all n-manifolds Mn such that Mn = Ul1 Ui, where UiC U+i and each Ui is a topological copy of U. The size of the collection is quite variable and we list some examples from which obvious questions arise. 1. If S2 is the 2-sphere and El the real line, let U=S2XE'. In this case 91Z( U) has two members, U and E3, where E3 is euclidean 3-space. 2. Let Mn be a combinatorial compact n-manifold with nF64. If U is the complement of a flat n-cell in Mn, then 1( U) has U as its only member [5]. The same result when U is a cell follows from [2]. 3. If U is S2 with an infinite convergent sequence of points removed, then Mlt(U) consists of exactly the proper open connected sets (proper domains) in S2. A manifold such as U in example 2 above for which 1Z(U) consists of one element is said to have the monotone union property. If Mn is an n-manifold, then a standard decomposition of Mn is a representation of Mn as a disjoint union, Mn= PUR, where P =En and dim R<n-1 [3].