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Boundary sets for growth hyperspaces. (English) Zbl 0627.54004
A subspace G of the hyperspace \(2^ X\) of the Peano continuum X is called a growth hyperspace if G contains every order arc A in \(2^ X\) such that \(\cap A\in G\). If G and H are growth hyperspaces of \(2^ X\) such that H is compact, G is a \(\sigma\)-compact dense subset of H, and the identity map on H can be approximated by continuous maps of H into H- G, then G is called a growth boundary set for H. It is known that H is homeomorphic to the Hilbert cube and that H-G is homeomorphic to Hilbert space. The authors investigate the existence and properties of growth boundary sets for certain growth hyperspaces. They are particularly interested in growth boundary sets which are cap sets or f-d cap sets. The definitions of these sets are too technical to be given here.
Reviewer: B.J.Pearson

MSC:
54B20 Hyperspaces in general topology
54F15 Continua and generalizations
57N20 Topology of infinite-dimensional manifolds
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[1] R.D. Anderson, On sigma-compact subsets of infinite-dimensional spaces, unpublished manuscript.
[2] Bessaga, C.; Pelczynski, A., The estimated extension theorem, homogeneous collections and skeletons, and their application to the topological classification of linear metric space s and convex sets, Fund. math., 69, 153-190, (1970) · Zbl 0204.12801
[3] Bessaga, C.; Pelczynski, A., Selected topics in infinite-dimensional topology, (1975), Polish Scientific Publishers Warsaw · Zbl 0304.57001
[4] Bing, R.H., Partitioning a set, Bull amer. math. soc., 55, 1101-1110, (1949) · Zbl 0036.11702
[5] Chapman, T.A., Dense sigma-compact subsets of infinite-dimensional manifolds, Trans. amer. math. soc., 154, 399-426, (1971) · Zbl 0208.51903
[6] Curtis, D.W.; Schori, R.M., Hyperspaces which characterize simple homotopy type, Gen. topology appl., 6, 153-165, (1976) · Zbl 0328.54003
[7] Curtis, D.W.; Schori, R.M., Hyperspaces of Peano continua are Hilbert cubes, Fund. math., 101, 19-38, (1978) · Zbl 0409.54044
[8] Curtis, D.W., Growth hyperspaces of Peano continua, Trans. amer. math. soc., 238, 271-283, (1978) · Zbl 0344.54009
[9] Curtis, D.W., Hyperspaces of noncompact metric spaces, Comp. math., 40, 139-152, (1980) · Zbl 0431.54004
[10] Curtis, D.W., Boundary sets in the Hilbert cube, Topology appl., 20, 201-221, (1985) · Zbl 0575.57008
[11] Curtis, D.W., Hyperspaces of finite subsets as boundary sets, Topology appl., 22, 97-107, (1986) · Zbl 0575.54009
[12] Henderson, J.P.; Walsh, J.J., Examples of cell-like decompositions of the infinite-dimensional manifolds σ and σ, Topology appl., 16, 143-154, (1983) · Zbl 0525.57011
[13] Kelley, J.L., Hyperspaces of a continuum, Trans. amer. math. soc., 52, 22-36, (1942) · Zbl 0061.40107
[14] Kroonenberg, N., Pseudo-interiors of hyperspaces, Comp. math., 32, 113-131, (1976) · Zbl 0336.54008
[15] Michael, M., Sigma-compact subsets of hyperspaces, ()
[16] Michael, M., Some hyperspaces homeomorphic to separable Hilbert space, (), 291-294
[17] Moise, E.E., Grille decomposition and convexification theorems for compact metric locally connected continua, Bull. amer. math. soc., 55, 1111-1121, (1949) · Zbl 0036.11801
[18] Torunczyk, H., Skeletonized sets in complete metric spaces and homeomorphisms of the Hilbert cube, Bull. acad. polon. sci., 18, 119-126, (1970) · Zbl 0202.54003
[19] Torunczyk, H., On CE-images of the Hilbert cube and characterization of Q-manifolds, Fund. math., 106, 31-40, (1980) · Zbl 0346.57004
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