Answering a question of Halbeisen we prove (by two different methods) that the algebraic dimension of each infinite-dimensional complete linear metric space X equals the size of X. A topological method gives a bit more: the algebraic dimension of a linear metric space X equals |X| provided the hyperspace K(X) of compact subsets of X is a Baire space. Studying the interplay between Baire properties of a linear metric space X and its hyperspace, we construct a hereditarily Baire linear metric space...

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 prove that, for any Hausdorff continuum X, if dim X ≥ 2 then the hyperspace C(X) of subcontinua of X is not a C-space; if dim X = 1 and X is hereditarily indecomposable then either dim C(X) = 2 or C(X) is not a C-space. This generalizes some results known for metric continua.

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.

A homeomorphism f:X → X of a compactum X with metric d is expansive if there is c > 0 such that if x, y ∈ X and x ≠ y, then there is an integer n ∈ ℤ such that $d(f^n\left(x\right),f^n\left(y\right))>c$. In this paper, we prove that if a homeomorphism f:X → X of a continuum X can be lifted to an onto map h:P → P of the pseudo-arc P, then f is not expansive. As a corollary, we prove that there are no expansive homeomorphisms on chainable continua. This is an affirmative answer to one of Williams’ conjectures.

We study the set functions 𝓣 and 𝒦 on irreducible continua. We present several properties of these functions when defined on irreducible continua. In particular, we characterize the class of irreducible continua for which these functions are continuous. We also characterize the class of 𝒦-symmetric irreducible continua.

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 $Con{v}_F\left(ℝⁿ\right)$ be the space of all non-empty closed convex sets in Euclidean space ℝ ⁿ endowed with the Fell topology. We prove that $Con{v}_F\left(ℝⁿ\right)\approx ℝⁿ\times Q$ for every n > 1 whereas $Con{v}_F\left(ℝ\right)\approx ℝ\times$.

Let X be an atom (= hereditarily indecomposable continuum). Define a metric ϱ on X by letting $\varrho (x,y)=W\left(A_{xy}\right)$ where $A_{x,y}$ is the (unique) minimal subcontinuum of X which contains x and y and W is a Whitney map on the set of subcontinua of X with W(X) = 1. We prove that ϱ is an ultrametric and the topology of (X,ϱ) is stronger than the original topology of X. The ϱ-closed balls C(x,r) = y ∈ X:ϱ ( x,y) ≤ r coincide with the subcontinua of X. (C(x,r) is the unique subcontinuum of X which contains x and has Whitney value...