Hyperspaces of Peano continua of euclidean spaces

If X is a space then L(X) denotes the subspace of C(X) consisting of all Peano (sub)continua. We prove that for n ≥ 3 the space L(R) is homeomorphic to B∞, where B denotes the pseudo-boundary of the Hilbert cube Q. Introduction. For a space X, C(X) denotes the hyperspace of all nonempty subcontinua of X. It is known that for a Peano continuum X without free arcs, C(X) ≈ Q, where Q denotes the Hilbert cube (Curtis and Schori [7]). L(X) denotes the subspace of C(X) consisting of all nonempty locally connected continua. The spaces L(X) were first studied by Kuratowski in [11]. He proved that L(X) is an Fσδ-subset of C(X), i.e., a countable intersection of σ-compact subsets. A little later, Mazurkiewicz [12] proved that for n ≥ 3, L(R) belongs to the Borel class Fσδ \Gδσ . Our main result is that for n ≥ 3 the spaces L(R) are homeomorphic to the countable infinite product of copies of the pseudo-boundary B of Q. Our methods do not apply to the case n = 2. We use the theory of absorbing sets in the Hilbert cube and some ideas from Dijkstra, van Mill and Mogilski [9]. In fact, we prove that for n ≥ 3, L([−1, 1]) is an Fσδ-absorber in C([−1, 1] ). Our main result then follows easily. We are indebted to R. Cauty for finding an inaccuracy in an earlier version of this manuscript. 1991 Mathematics Subject Classification: Primary 57N20.