Hyperspaces of finite subsets which are homeomorphic to ℵ0-dimensional linear metric spaces

Abstract

We prove that the hyperspace of closed bounded sets with the Hausdorff topology, over an almost convex metric space, is an absolute retract. Dense subspaces of normed linear spaces are examples of, not necessarily connected, almost convex metric spaces. We give some necessary conditions for the path-wise connectedness of the Hausdorff metric topology on closed bounded sets. Finally, we describe properties of a separable metric space, under which its hyperspace with the Wijsman topology is path-wise connected.

Finite subset spaces of graphs and punctured surfaces

“…Other authors including Curtis [8], Curtis and To Nhu [9], Handel [11], Illanes [12] and Macías [14] have established general topological and homotopytheoretic properties of exp k X and exp X , and Beilinson and Drinfeld [2, sec. 3.5.1] and Ran [17] have used these spaces in the context of mathematical physics and algebraic geometry.…”

Stable homology of automorphism groups of free groups

“…It seems reasonable that this map should be a homotopy equivalence (it even seems close to being a homeomorphism: If we had considered instead piecewise linear graphs, it would be a bijection). Curtis and To Nhu [CTN85] proves that Sub(S N −1 ) is contractible. (In fact they prove that it is homeomorphic to R ∞ .…”

Characterizing the topology of infinite-dimensional 𝜎-compact manifolds

“…Let us note that for er-compact ANR's (7) is equivalent to (5). Details of proofs and related examples will appear in [21].…”

“…Several natural pairs of infinite-dimensional spaces have a structure of (¿2, Zf ^manifolds (cf. [3,5,8,10]). To recognize them the following characterization was elaborated (cf.…”

“…Therefore the map h = limpnhn is a homeomorphism of X onto Y such that d(f(x),h(x)) < 1 for all x eX. 5. Characterization of V2-and E-manifolds.…”