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Ordered embeddings of symmetric products. (English) Zbl 1151.54011

In this admirably well-presented paper, the authors give clear and simple results which increase our understanding of the relationship of order and hyperspaces \(F_n(X)\) of non-empty closed subsets of metric continua \(X\), which contain at most \(n\) elements. They say that \(F_n(X)\) can be orderly embedded in \(F_m(Y)\) provided there exists an embedding \(h: F_n(X)\to F_m(Y)\) such that if \(A,B\in F_n(X)\) and \(A\subseteq B\), then \(h(A)\subseteq h(B)\). They prove that (a) if \(n\leq m< 2n\) and \(F_n(X)\) can be orderly embedded in \(F_m(T)\), then \(X\) can be embedded in \(Y\), and (b) there exist continua \(X\), \(Y\) such that, for every \(n\geq 1\), \(F_n(X)\) can be orderly embedded in \(F_{2n}(Y)\) and \(X\) cannot be embedded in \(Y\).

MSC:

54B20 Hyperspaces in general topology