Boundary sets for growth hyperspaces

AbstractA subspace G of the hyperspace 2X of a Peano continuum is called a growth hyperspace if G contains every order arc α in 2X for which ∩ α ∈ G. If G ⊂ H ⊂ 2X are growth hyperspaces such that H is compact, G is σ-compact and dense in H, and the identity map on H is approximable by maps into H ⧹ G, then H is homeomorphic to the Hilbert cube and H ⧹ G is homeomorphic to Hilbert space, and we call G a growth boundary set for H. We investigate the existence and the nature of growth boundary sets for growth hyperspaces of the types 2X(A)={F∈2X:F⌢A≠∅} and C(X;A)={M∈2X:M is a continuum and M⌢A≠∅}, where A ∈ 2X. In parti cular, we characterize those situations in which there exist growth boundary sets which are f-d cap sets, and those situations in which every growth boundary set is a cap set