Ordered embeddings of symmetric products

Abstract

Let X and Y be metric continua. Let F-n(X) (resp., F-n(Y)) be the hyperspace of nonempty closed subsets of X (resp., Y) which contain at most n elements. We say that the hyperspace F-n(X) can be orderly embedded in F-m (Y) provided that there exists an embedding h : F-n (X) -> F-m (Y) such that if A, B E Fn (X) and A C B, then h(A) C h(B). In this paper we prove: (a) If n <= m < 2n and F-n (X) can be orderly embedded in F-m (Y), then X can be embedded in Y. (b) There exist continua X and Y such that, for each n >= 1, F-n (X) can be orderly embedded in F-2n (Y) and X can not be embedded in Y.