Embedding products of chainable continua

Abstract

Bing [1 ] showed that every chainable continuum can be embedded in E2. A consequence of this is that if each of A1, * * *, A,. is a chainable continuum, then the topological product A1X * * * XA. can be embedded in E2n. This fact can also be derived from a theorem of Isbell [2]. This paper shows that the integer 2n can be replaced by n+ 1. The following example shows it cannot be replaced by n. EXAMPLE. For each integer n larger than 1 the product of n-I arcs and a sin(1/x) curve cannot be embedded in En. Such a continuum contains an n-cell and a subset disjoint from the n-cell with limit points in the interior of the n-cell. McCord [4] has proved an elegant embedding theorem which will be the principal tool used here. He defines a map f from a compact subset X of a metric space (E, d) to a compact subset Y of E to be approximable by homeomorphisms (relative to E) provided that for every e> 0 there is an open set U containing X and a 1 to 1 map ,g of U into E such that for all x in X, d(u(x), f(x)) <e.