Hyperspaces of dimension 1

More absorbers in hyperspaces

The family of all subcontinua that separate a compact connected $n$-manifold $X$ (with or without boundary), $n\ge 3$, is an $F_σ$-absorber in the hyperspace $C(X)$ of nonempty subcontinua of $X$. If $D_2(F_σ)$ is the small Borel class of spaces which are differences of two $σ$-compact sets, then the family of all $(n-1)$-dimensional continua that separate $X$ is a $D_2(F_σ)$-absorber in $C(X)$. The families of nondegenerate colocally connected or aposyndetic continua in $I^n$ and of at least two-dimensional or decomposable Kelley continua are $F_{σδ}$-absorbers in the hyperspace $C(I^n)$ for …

The dimension of hyperspaces of non-metrizable continua

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

关于纽结与平面简单闭曲线的超空间

On hyperspaces of knots and planar simple closed curves

本文研究了 $\mathbb R^n$ 中简单闭曲线与纽结的 Vietoris 超空间，证明了这些超空间具有强局部可缩、弧连通及无穷维 Cantor 流形等拓扑性质，并阐明了特定子空间的绝对邻域收缩核属性。

The topology of systems of hyperspaces determined by dimension functions

Given a non-degenerate Peano continuum $X$, a dimension function $D:2^X_*\to[0,\infty]$ defined on the family $2^X_*$ of compact subsets of $X$, and a subset $Γ\subset[0,\infty)$, we recognize the topological structure of the system $(2^X,\D_{\leγ}(X))_{α\inΓ}$, where $2^X$ is the hyperspace of non-empty compact subsets of $X$ and $D_{\leγ}(X)$ is the subspace of $2^X$, consisting of non-empty compact subsets $K\subset X$ with $D(K)\leγ$.

Hyperspaces of countable compacta

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 ($n=ω$) or countably many ($n=ω+1$) accumulation points are studied. By descriptive set-theoretical methods, we fully characterize them for 0-dimensional, dense-in-itself, Polish spaces and partially for $σ$-compact spaces $X$. Using the theory of absorbing sets, we get characterizations of $\mathcal H(X)$, $\mathcal A_ω(X)$ and $\mathcal A_{ω+1}(X)$ for nondegenerate connected, locally connected Polish s…

k维闭凸集超空间

The hyperspace of k-dimensional closed convex sets

本文研究欧几里得空间中k维闭凸集超空间的拓扑结构，证明其与紧凸子集超空间均为Hilbert方体流形，并具有Grassmann流形上的纤维丛结构，同时确定了紧凸情形纤维的同胚类型。