Regular Convergence of Manifolds with Boundary

In the author's paper [l ] it is shown that if a sequence of orientable »-dimensional generalized closed manifolds (abbreviated »-gem) converge (« —1)-regularly to an »-dimensional set, then the limit set is also an orientable »-gem (see Definitions 1 and 2). In this paper a similar result is obtained for manifolds with boundary. Throughout the paper we assume that our sets are imbedded in a compact Hausdorff space S and that the cycles are Cech cycles with coefficients in an arbitrary field which we will omit from our notation for a cycle. All of the basic homology theory needed is in [7] and a knowledge of it will be assumed. We shall use the notation {^4,}—*A to mean that the sequence of sets ^4»CS converges to A ES as a limit (see p. 10 of [S]). We shall consider convergence only when all of the sets Ai are closed, and shall not explicitly state this henceforth; the limit set, as is well known, will always be closed. We shall use small latin letters for points, large ones for sets, and script letters for finite collections of open sets which are coverings of S. We shall use "\J" for point set union or sum, "Hi" for intersection, reserving + and — for the group operations. By V/\A we shall mean the subcomplex of the nerve of 1) whose vertices are elements of the covering V that have a nonvacuous intersection with A. If every element of U is contained in some element of V, we shall say that V is a refinement of V, and shall write TJ>V, or 1) = 1)(\r*); we shall use the notation1 Hvu to denote a simplicial projection from the nerve of V into the nerve of V. By U C C V we mean that UEV, and by TJ^>V we mean that the closure of the elements of V form a refinement ofV, and by V> *V, we mean that the stars of the elements of V form a refinement of V. If a cell a of a covering V has a nucleus (the intersection of all the open sets representing the vertices of a) that meets a set B, we say that a is on B; If all the cells with nonzero coefficients in a chain C of V are on B, we say that Cr is on B. If the nucleus of <r is in the open set Q, we say a is in Q. We shall use the symbol "3" to denote the boundary operator. Definition 1. We say that {Ai}—*A n-regularly if and only if {Ai}-^A and corresponding to any coverings T^V, there is a covering Vn "Vn (V) and an integer N each depending only on V, and a cover-