We characterize metric spaces whose hyperspaces of non-empty closed, bounded, compact or finite subsets, endowed with the Attouch-Wets topology, are absolute (neighborhood) retracts.

Let $Con{v}_H\left(X\right)$, $Con{v}_{AW}\left(X\right)$ and $Con{v}_W\left(X\right)$ be the spaces of all non-empty closed convex sets in a normed linear space X admitting the Hausdorff metric topology, the Attouch-Wets topology and the Wijsman topology, respectively. We show that every component of $Con{v}_H\left(X\right)$ and the space $Con{v}_{AW}\left(X\right)$ are AR. In case X is separable, $Con{v}_W\left(X\right)$ is locally path-connected.

We consider separable metrizable topological spaces. Among other things we prove that there exists a non-contractible space with the compact extension property and we prove a new version of realization of polytopes for $ANR$’s.

We prove that if G is a locally compact Hausdorff group then every proper G-ANR space has the G-homotopy type of a G-CW complex. This is applied to extend the James-Segal G-homotopy equivalence theorem to the case of arbitrary locally compact proper group actions.

The aim of this paper is to prove the generalized Schoenflies theorem for the class of absolute suspensions. The question whether the finite-dimensional absolute suspensions are homeomorphic to spheres remains open. Partial solution to this question was obtained in [Sz] and [Mi]. Morton Brown gave in [Br] an ingenious proof of the generalized Schoenflies theorem. Careful analysis of his proof reveals that modulo some technical adjustments a similar argument gives an analogous result for the class...

It is shown that the hyperspace of non-empty finite subsets of a space X is an ANR (an AR) for stratifiable spaces if and only if X is a 2-hyper-locally-connected (and connected) stratifiable space.

This paper discusses the notion, the properties and the application of multicores, i.e. some compact sets contained in metric spaces.

The hyperspaces $ANR\left({ℝ}^n\right)$ and $AR\left({ℝ}^n\right)$ in $2^{{ℝ}^n}\left(n\ge 3\right)$ consisting respectively of all compact absolute neighborhood retracts and all compact absolute retracts are studied. It is shown that both have the Borel type of absolute $G_{\delta \sigma \delta }$-spaces and that, indeed, they are not $F_{\sigma \delta \sigma }$-spaces. The main result is that $ANR\left({ℝ}^n\right)$ is an absorber for the class of all absolute $G_{\delta \sigma \delta }$-spaces and is therefore homeomorphic to the standard model space ${\Omega }_3$ of this class.

Let J(n) be the hyperspace of all centrally symmetric compact convex bodies $A\subseteq {ℝ}^n$, n ≥ 2, for which the ordinary Euclidean unit ball is the ellipsoid of maximal volume contained in A (the John ellipsoid). Let $J_0\left(n\right)$ be the complement of the unique O(n)-fixed point in J(n). We prove that: (1) the Banach-Mazur compactum BM(n) is homeomorphic to the orbit space J(n)/O(n) of the natural action of the orthogonal group O(n) on J(n); (2) J(n) is an O(n)-AR; (3) $J_0\left(2\right)/SO\left(2\right)$ is an Eilenberg-MacLane space $𝐊(\mathbb{Q},2)$; (4) $B{M}_0\left(2\right)=J_0\left(2\right)/O\left(2\right)$ is noncontractible;...

Let X be a topological group or a convex set in a linear metric space. We prove that X is homeomorphic to (a manifold modeled on) an infinite-dimensional Hilbert space if and only if X is a completely metrizable absolute (neighborhood) retract with ω-LFAP, the countable locally finite approximation property. The latter means that for any open cover $$𝒰$$ of X there is a sequence of maps (f n: X → X)nεgw such that each f n is $$𝒰$$ -near to the identity map of X and the family f n(X)n∈ω is locally finite...