On the hyperspace of subcontinua of an arc-like continuum

Abstract

It is shown that the hyperspace of each arc-like continuum can be embedded in W. R. R. Transue [1 ] in a beautiful note gave a positive answer to A. Connor's [2, p. 152 ] question Can the hyperspace of subcontinua of the pseudoarc (with the Hausdorff metric) be embedded in E3. This note extends this result to arc-like continua, i.e. inverse limits on arcs originally called snake-like continua by R. H. Bing [3 ]. Here { W, fi I will denote the inverse limit system with indexing set the nonnegative integers and with each factor space W. The associated inverse limit space will be denoted by lim { W, fi }I. See [4, p. 87 ] for a discussion of inverse limits. The hyperspace of continua of a space X, denoted by C(X), is studied in [5 ]. The closed interval [0, 1] will be called I. THEOREM. The hyperspace of continua of the inverse limit space X = lim { I, fi I embeds in PROOF. There is no loss to assume that none of the maps fi is constant on an open set. Since a continuum in I is either a closed interval or a point, C(I) will be identified with D = {(x, y, z) C E310 < x < y < 1, z = O. Take Fi:D-*D by Fi(x, y, 0) = (minfi(t), maxfi(t), 0), t E [x, y]. Now Fi is the natural map from C(I) to C(I) induced by fi. J. Segal [6] proved that the hyperspace of continua of the inverse limit space X is homeomorphic to lim{ C(I), F. }. The proof of the theorem will be completed by embedding lim {D, Fi I in Now if each of the maps Fi could be approximated by embeddings in E3 in the Received by the editors December 8, 1969. AMS 1970 subject classifications. Primary 54B20, 54C25, 54F50.