Theorem 1 answers a question posed by Wojdyslawski [8], who later showed that 2 is an AR for every Peano space X [9], The converse is easily seen to be true; in fact, if 2 is locally connected, then so is X. The proof of Theorem 1 is based on the recent result of Schori and West [5] that 2£&Q for every nondegenerate connected graph Y. Wojdyslawski also showed that C(X) is an AR if (and only if) X is a Peano space. An important special case of Theorem 2 is already known: if T is a connected graph, then C(Y) is a contractible polyhedron [4], and therefore C(Y)xQ^Q by a theorem of West [6]. Since C(I)^I\ the condition that X contains no free arcs is clearly necessary for C(X)^Q. The proof of sufficiency uses a recent result of West [7] that C(D)^Q for every dendron D with a dense set of branch points.

A function ƒ:Rn→Rn, n⩾2, is sectionally continuous if each restriction ƒ|H to an (n − 1)-dimensional hyperplane H is continuous. We show that a sectionally continuous injection ƒ is continuous at a point x in Rn if and only if ƒ(x) is not a limit point of any component of Rnƒ(Rn). In particular, ƒ is an imbedding if and only if ƒ(Rn) is open. For n = 2, we also describe all possible images for sectionally continuous injections with only countably many discontinuities.

For a tower X 1 ⊂ X 2 ⊂ ⋯ of locally compact metric spaces, let X ∞ = ∪ ∞ 1 X n denote the direct limit space. We show that the hyperspace 2 X ∞ of nonempty compact subsets of X ∞ , with the Vietoris topology, is homeomorphic to the direct limit of the tower of hyperspaces 2 X 1 ∪2 X 2 ∪⋯. Consequently, if each X n is a generalized Peano continuum, with X n closed and nowhere dense in X n +1 , then 2 X ∞ is homeomorphic to the direct limit of Hilbert cubes.

Let X denote a connected, locally path-connected, σ-compact metric space. F(X) is the hyperspace of all nonempty finite subsets of X, topologized by the Hausdorff metric. Let E denote a σ-compact subspace of F(X) with the property that, for EϵE and FϵF(X) with E⋐F, FϵE. If X admits a Peano compactification X̄, then E is a σZ-set in its closure Ē in the hyperspace 2x̄, and Ē is a topological Hilbert cube. We show that E contains an fd-cap set (and is therefore a boundary set) for Ē if and only if the remainder X̄\X is locally non-separating in X̄. In particular, if X=X̄ is a Peano continuum, then F(X) is a boundary set for 2x.

We study properties of those σZ-sets in the Hilbert cube whose complements are homeomorphic to Hilbert space. A characterization of such sets is obtained in terms of a proximate local connectedness property and a dense imbedding condition. Some examples and applications are given, including the formulation of a tower condition useful for recognizing (f-d) cap sets.

For X a metric continuum, 2X denotes the hyper space of all nonempty subcompacta, with the topology induced by the Hausdorff metric H, and C(X) ⊂ 2X the hyperspace of subcontinua. These hyperspaces are continua, in fact are arcwise-connected, since there exist order arcs between each hyperspace element and the element X. They also have trivial shape, i.e., maps of the hyperspaces into ANRs are homotopic to constant maps. For a detailed discussion of these and other general hyperspace properties, we refer the reader to Nadler's monograph [4].The question of hyperspace contractibility was first considered by Wojdyslawski [8], who showed that 2X and C(X) are contractible if X is locally connected. Kelley [2] gave a more general condition (now called property K) which is sufficient, but not necessary, for hyperspace contractibility. The continuum X has property K if for every there exists δ > 0 such that, for every pair of points x, y with d(x, y) < δ and every subcontinuum M containing x, there exists a subcontinuum N containing y with .

It is shown that if H is a connected, locally contractible, separable, topologically complete metric space with the property that mappings of separable metric spaces into H are approximable by imbeddings (in particular, if H is Hilbert space), then every sigma-compact, nowhere locally compact metric space can be densely imbedded in H.

We give an example of a topologically complete separable metric AR space $X$ which is not homeomorphic to the Hilbert space ${l^2}$, but which has the following properties: (i) $X$ imbeds as a convex subset of ${l^2}$ (ii) every compact subset of $X$ is a $Z$-set; (iii) $X \times X \approx {l^2};$ (iv) $X$ is homogeneous; (v) $X \approx X\backslash G$ for every countable subset $G$.

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For X a nondegenerate Peano continuum, let 2 X {2^X} be the hyperspace of all nonempty closed subsets of X, topologized with the Hausdorff metric. It is known that 2 X {2^X} is homeomorphic to the Hilbert cube. A nonempty closed subspace G \mathcal {G} of 2 X {2^X} is called a growth hyperspace provided it satisfies the following condition: if A ∈ G A \in \mathcal {G} , and B ∈ 2 X B \in {2^X} such that B ⊃ A B \supset A and each component of B meets A, then also B ∈ G B \in \mathcal {G} . The class of growth hyperspaces includes many previously considered subspaces of 2 X {2^X} . It is shown that if X contains no free arcs, and G \mathcal {G} is a nontrivial growth hyperspace, then G ∖ { X } \mathcal {G}\backslash \{ X\} is a Hilbert cube manifold. A corollary characterizes those growth hyperspaces which are homeomorphic to the Hilbert cube. Analogous results are obtained for growth hyperspaces with respect to the hyperspace cc ( X ) {\text {cc}}(X) of closed convex subsets of a convex n-cell X.

A brick decomposition (respectively, generalized brick decomposition) of a metric space Y is a locally finite, star-finite closed cover { Y α } \{ {Y_\alpha }\} such that each nonempty intersection Y α 1 ∩ ⋯ ∩ Y α n , n ⩾ 1 {Y_{{\alpha _1}}} \cap \cdots \cap {Y_{{\alpha _n}}},n \geqslant 1 , is a compact AR (respectively, locally compact AR). Let K be the nerve of the decomposition { Y α } \{ {Y_\alpha }\} , let Q be the Hilbert cube, and Q 0 = Q ∖ point ≈ Q × [ 0 , 1 ) {Q_0} = Q\backslash \;\text {point}\approx Q \times [0,1) . Then Y × Q ≈ | K | × Q Y \times Q \approx |K| \times Q (respectively, Y × Q 0 ≈ | K | × Q 0 Y \times {Q_0} \approx |K| \times {Q_0} ).

The star hyperspace 2KSSt of a non-degenerate finite connected simplicial complex K is defined as the space of nonempty closed subsets of |K| that lie in small neighborhoods of the vertex stars of K (with respect to a barycentric subdivision), topologized with the Hausdorff metric. It has previously been shown that the hyperspace 2X of all nonempty closed subsets of a non-degenerate Peano space X is homeomorphic to the Hilbert cube Q. In this paper it is shown that 2KSSt is a Q-manifold homeomorphic to |K| × Q. It follows from a result of T.A. Chapman that complexes K and L have the same simple homotopy type if and only if 2KSSt and 2LSSt are homeomorphic. A main tool in studying star hyperspaces is the following result, also obtained in this paper: if X is a non-degenerate Peano space and A1, …,Anϵ2X, then the space of closed subsets of X intersecting each Ai is homeomorphic to Q.