Cells and Cellularity in Infinite-Dimensional Normed Linear Spaces

Abstract

Certain concepts such as cells, cellular sets, point-like sets, and decomposition spaces are studied and related in normed linear spaces. The relationships between these concepts in general resemble somewhat the corresponding relationships in Euclidean space. There are certain topological properties in Euclidean n-space which can be conveniently studied as properties in normed linear spaces. In this paper con, cepts such as open and closed cells, cellular sets, point-like sets, and decomposition spaces are studied and related. Many, but not all, of the relationships between these concepts in infinite-dimensional normed linear spaces resemble the corresponding relationships in finite-dimensional spaces. Throughout the paper, E will denote an arbitrary normed linear space, and 0 will represent the zero element of E. For any positive real number r and any x ,E E let B r(x) =y 6 E: llx Y|| < r} and Sr(x) -y e E: l|x Y|| = r}. For convenience let Br Br(0) and Sr = Sr(0). 1. Tame cells. A closed subset C of E is a cell in E if there exists a homeomorphism from the pair (B1, SI) onto the pair (C, Bd C). C is tame if there exists a homeomorphism from E onto itself taking C onto Bl. A closed subset K of E\Int C is a collar of C if there exists a homeomorphism h from the triple (B1; B2 $SI) onto the triple (K u C; C, Bd (K u C)). Lemma 1.1. Let C be a cell in E, and let / be a homeomorphism /rom the pair (BP, S onto the pair (C, Bd C). Then there exists a homeomorphism h from E onto itself such that h B = fIB V2 ./2 Presented to the Society, April 20, 1968; received by the editors September 13, 1971 and, in revised form, February 23, 1972. AMS (MOS) subject classifications (1970). Primary 57A17, 57A60; Secondary 57A20.