A necessary condition that a cellular upper semi-continuous decomposition of 𝐸ⁿ yield 𝐸ⁿ
T. M. Price · Transactions of the American Mathematical Society · 1966
Mathematical Society 10 (1963), 661 for decompositions of £3.The proof has since been simplified and found to hold in £" for n = 5 as well.As is consistent with the present seeming lack of knowledge of £4, Theorem 2.2 is not known to be true in £4.Its validity in E1 and £2 follows from already well known results.Let upper semicontinuous be defined as in [2].A decomposition of £", G, is called monotone if each element of G is connected.It is called cellular if each element is the intersection of a decreasing sequence of «-cells.The letter G will be used to denote both the collection of subsets of £"and the resulting decomposition space.The letter H will be used to denote the collection of nondegenerate elements of G, and H* will denote the union of the elements of H.A space X is called 1-connected at infinity if for each compact subset C of X there is a compact subset B of X such that C £ B and X -B is connected and simply connected.In [5] Edwards defined a triangulated 3-manifold, M, to be 1-connected at infinity if for each compact subset A of M there exists a compact polyhedral subset P of M such that A £ p and M -P is 1-connected.It is easy to show, using the techniques described by D. R. McMillan, Jr. in [6] that if B is a compact subset of a triangulated 3-manifold, M, and if M -B is 1-connected, then there exists a compact polyhedral subset P of M such that B s p and M -P is 1-connected.Hence the two definitions are equivalent in 3-manifolds.The definitions of piecewise linear concepts used are as given in [7].If B is an n-cell, Pis used to denote the boundary of B.1.A sufficient condition that a decomposition of /-.'" yield £" as the decomposition space.Theorem 1.4 states a condition which is sufficient to insure that the decomposition space is again E".The condition is also shown to be necessary for countable decompositions of E3.Unfortunately, the sufficient condition given in