Florencio Corona-Vázquez, Russell Aarón Quiñones-Estrella, Javier Sánchez Martínez
Let $X$ be a continuum and let $C(X)$ denote the hyperspace of subcontinua of $X$, endowed with the Hausdorff metric. For $p\in X$, define the hyperspace $C(p,X)=\{A\in C(X):p\in A\}$ as a subspace of $C(X)$. In this paper we introduced the quotient space $HS(p,X)=C(X)/C(p,X)$. We present some general properties of $HS(p,X)$ and we study the relationship between the continuum $X$ and the hyperspaces $C(X)$ and $HS(p,X)$.
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The hyperspace of non-cut subcontinua of graphs
本文研究了连续统的非割子连续统超空间$NC^{*}(X)$，证明了使其紧致的局部连通连续统仅为弧和简单闭曲线，并刻画了使该超空间连通的有限图。
The hyperspace of non-cut subcontinua of graphs and dendrites
研究有限图与树形的非割子连续统超空间 $NC^{*}(X)$，给出其紧性、连通性及全不连通的条件，并证明端点集稠密的树形对应超空间同胚于无理数的 Baire 空间。
Let $X$ be a nondegenerate Peano unicoherent continuum. The family $CB(X)$ of proper subcontinua of $X$ with connected boundaries is a $G_δ$-subset of the hyperspace $C(X)$ of all subcontinua of $X$. If every nonempty open subset of $X$ contains an open subset homeomorphic to $\mathbb R^n$ (such space is called $π$-$n$-Euclidean) and $2\le n<\infty$, then $C(X)\setminus CB(X)$ is recognized as an $F_σ$-absorber in $C(X)$; if additionally, no one-dimensional subset separates $X$, then the family of all members of $CB(X)$ which separate $X$ is a $D_2(F_σ)$-absorber in $C(X)$, where $D_2(F_σ)$ de…
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 …
It is shown that the family of all homogeneous continua in the hyperspace of all subcontinua of any finite-dimensional Euclidean cube or the Hilbert cube is an analytic subspace of the hyperspace which contains a topological copy of the linear space $c_0=\{(x_k)\in \mathbb R^ω: \lim x_k=0\}$ as a closed subset.
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.
Given a continuum $X$ and $p\in X$, we will consider the hyperspace $C(p,X)$ of all subcontinua of $X$ containing $p$. Given a family of continua $\mathcal{C}$, a continuum $X\in\mathcal{C}$ and $p\in X$, we say that $(X,p)$ has unique hyperspace $C(p,X)$ relative to $\mathcal{C}$ if for each $Y\in\mathcal{C}$ and $q\in Y$ such that $C(p,X)$ and $C(q,Y)$ are homeomorphic, then there is an homeomorphism between $X$ and $Y$ sending $p$ to $q$. In this paper we show that $(X,p)$ has unique hyperspace $C(p,X)$ relative to the classes of dendrites if and only if $X$ is a tree, we present also some …
Given a continuum $X$ and $p\in X$, we will consider the hyperspace $C(p,X)$ of all subcontinua of $X$ containing $p$. Given a family of continua $\mathcal{C}$, a continuum $X\in\mathcal{C}$ and $p\in X$, we say that $(X,p)$ has unique hyperspace $C(p,X)$ relative to $\mathcal{C}$ if for each $Y\in\mathcal{C}$ and $q\in Y$ such that $C(p,X)$ and $C(q,Y)$ are homeomorphic, then there is an homeomorphism between $X$ and $Y$ sending $p$ to $q$. In this paper we study some topological and geometric properties about the structure of $C(p,X)$ when $X$ is a tree, being the main result that $(X,p)$ ha…
A continuum is a compact connected metric space. A non-empty closed subset $B$ of a continuum $X$ does not block $x\in X\setminus B$ provided that the union of all subcontinua of $X$ containing $x$ and contained in $X\setminus B$ is dense in $X$. We denote the collection of all non-empty closed subset $B$ of $X$ such that $B$ does not block each element of $X\setminus B$ by $NB(F_1(X))$. In this paper we show some properties of the hyperspace $NB(F_1(X))$. Particularly, we prove that the simple closed curve is the unique continuum $X$ such that $NB(F_1(X))=F_1(X)$, given a positive answer to a…
Given a metric continuum $X$, a nonempty proper closed subspace $B$ of $X$, does not block a point $p\in X\setminus B$ provided that the union of all subcontinua of $X$ containing $p$ and contained in $X\setminus B$ is a dense subset of $X$. The collection of all nonempty proper closed subspaces $B$ of $X$ such that $B$ does not block any element of $X\setminus B$ is denoted by $NB(F_{1}(X))$. In this paper we prove that for each completely metrizable and separable space $Z$, there exists a continuum $X$ such that $Z$ is homeomorphic to $NB(F_{1}(X))$. This answers a series of questions by Cam…
Let X be a non-degenerate, connected, locally path-connected metrizable space and Fin(X) be the hyperspace consisting of non-empty finite subsets in X endowed with the Vietoris topology. In this paper, we show that every compact set in Fin(X) is a strong Z-set.
Let ${\rm Fin}(X)$ be the hyperspace consisting of non-empty finite subsets of a space $X$ endowed with the Vietoris topology. In this paper, we characterize a metrizable space $X$ whose hyperspace ${\rm Fin}(X)$ is homeomorphic to the linear subspace spanned by the canonical orthonormal basis of a non-separable Hilbert space.
Let $X$ be a metrizable space and ${\rm Comp}(X)$ be the hyperspace consisting of non-empty compact subsets of $X$ endowed with the Vietoris topology. In this paper, we give a necessary and sufficient condition on $X$ for ${\rm Comp}(X)$ to be homeomorphic to a non-separable Hilbert space. Moreover, we consider the topological structure of pair $({\rm Comp}(\overline{X}),{\rm Fin}(X))$ of hyperspaces of $X$ and its completion $\overline{X}$, where ${\rm Fin}(X)$ is the hyperspace of non-empty finite sets in $X$.
Given uniformly homeomorphic metric spaces $X$ and $Y$, it is proved that the hyperspaces $C(X)$ and $C(Y)$ are uniformly homeomorphic, where $C(X)$ denotes the collection of all nonempty closed subsets of $X$, and is endowed with Hausdorff distance. Gerald Beer has proved that the hyperspace $C(X)$ is Atsuji when $X$ is either compact or uniformly discrete. An Atsuji space is a generalization of compact metric spaces as well as of uniformly discrete spaces. In this article, we investigate the space $C(X)$ when $X$ is Atsuji, and a class of Atsuji subspaces of $C(X)$ is obtained. Using the obt…
We characterize metric spaces $X$ whose hyperspaces $2^X$ or $Bd(X)$ of non-empty closed (bounded) subsets, endowed with the Hausdorff metric, are absolute [neighborhood] retracts.
Let $CLB_H(X)$ denote the hyperspace of closed bounded subsets of a metric space $X$, endowed with the Hausdorff metric topology. We prove, among others, that natural dense subspaces of $CLB_H(R^m)$ of all nowhere dense closed sets, of all perfect sets, of all Cantor sets and of all Lebesgue measure zero sets are homeomorphic to the Hilbert space $\ell_2$. Moreover, we investigate the hyperspace $CL_H(R)$ of all nonempty closed subsets of the real line $R$ with the Hausdorff (infinite-valued) metric. We show that a nonseparable component of $CL_H(R)$ is homeomorphic to the Hilbert space $\ell_…
For a metric space $X$, let $\mathsf FX$ be the space of all nonempty finite subsets of $X$ endowed with the largest metric $d^1_{\mathsf FX}$ such that for every $n\in\mathbb N$ the map $X^n\to\mathsf FX$, $(x_1,\dots,x_n)\mapsto \{x_1,\dots,x_n\}$, is non-expanding with respect to the $\ell^1$-metric on $X^n$. We study the completion of the metric space $\mathsf F^1\!X=(\mathsf FX,d^1_{\mathsf FX})$ and prove that it coincides with the space $\mathsf Z^1\!X$ of nonempty compact subsets of $X$ that have zero length (defined with the help of graphs). We prove that each subset of zero length in…
Let G be a compact group acting on a Banach space L by means of linear isometries. The action of G on L induces a natural continuous action on cc(L), the hyperspace of all compact convex subsets of L endowed with the Hausdorff metric topology. The main result of this paper states that the G-space cc(L) is a G-AE. Under some extra assumptions, this result can be extended to CB(L), the hyperspace of all closed and bounded convex subsets of L.
There exists a completely metrizable bounded metrizable space $X$ with compatible metrics $d,d'$ so that the hyperspace $CL(X)$ of nonempty closed subsets of $X$ endowed with the Hausdorff metric $H_d$, $H_{d'}$, resp. is $α$-favorable, $β$-favorable, resp. in the strong Choquet game. In particular, there exists a completely metrizable bounded metric space $(X,d)$ such that $(CL(X),H_d)$ is not completely metrizable.
The hyperspace ω(f) when f is a transitive dendrite mapping
本文研究紧度量空间上连续映射诱导的 omega 极限集超空间 ω(f) 的拓扑性质，证明其内部为空，并在传递树枝映射下给出全不连通性条件及 Wazewski 万有树枝上存在含弧反例。
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。
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