Paweł Krupski

We consider the Vietoris hyperspaces $\mathcal S(\mathbb R^n)$ of simple closed curves in $\mathbb R^n$, $n=2,3$, and their subspaces $\mathcal S_P(\mathbb R^2)$ of planar simple closed polygons, $\mathcal K_P$ of polygonal knots, and $\mathcal K_T$ of tame knots. We prove that all the hyperspaces are strongly locally contractible, arcwise connected, infinite-dimensional Cantor manifolds, and $\mathcal S(\mathbb R^2)$ and $\mathcal K_T$ are strongly infinite-dimensional Cantor manifolds. Moreover, $\mathcal S_P(\mathbb R^2)$ and $\mathcal K_P$ are $σ$-compact, strongly countable-dimensional absolute neighborhood retracts.

The notion of a $V^n$-continuum was introduced by Alexandroff \cite{ps} as a generalization of the concept of $n$-manifold. In this note we consider the cohomological analogue of $V^n$-continuum and prove that any strongly locally homogeneous generalized continuum $X$ with cohomological dimension $\dim_G X=n$ is a generalized $V^n$-space with respect to the cohomological dimension $\dim_G$. In particular, every strongly locally homogeneous continuum of covering dimension $n$ is a $V^n$-continuum in the sense of Alexandroff. This provides a partial answer to a question raised in \cite{tv}. An analog of the Mazurkiewicz theorem that no subset of covering dimension $\le n-2$ cuts any region of the Euclidean $n$-space is also obtained for strongly locally homogeneous generalized continua $X$ of cohomological dimension $\dim_G X=n$.

Hyperspaces $\mathcal H(X)$ of all countable compact subsets of a metric space $X$ and $\mathcal A_n(X)$ of infinite compact subsets which have at most $n$ ($n\in\mathbb N$), or finitely many ($n=\omega$) or countably many ($n=\omega+1$) accumulation points are studied. By descriptive set-theoretical methods, we fully characterize them for 0-dimensional, dense-in-itself, Polish spaces and partially for $\sigma$-compact spaces $X$. Using the theory of absorbing sets, we get characterizations of $\mathcal H(X)$, $\mathcal A_\omega(X)$ and $\mathcal A_{\omega+1}(X)$ for nondegenerate connected, locally connected Polish spaces $X$ which are either locally compact or nowhere locally compact. For every $n\in\mathbb N$, we show that if $X$ is an interval or a simple closed curve, $\mathcal A_n(X)$ is homeomorphic to the linear space $c_{0}=\{(x_{i}) \in\mathbb R^{\omega}: \lim x_{i}=0\}$ with the product topology; if $X$ is a Peano continuum and a point $p\in X$ is of order $\ge 2$, then the hyperspace $\mathcal A_1(X,\{p\})$ of all compacta with exactly one accumulation point $p$ also is homeomorphic to $c_{0}$.