Wijsman Hyperspaces: Subspaces and Embeddings
Jiling Cao, Heikki J. K. Junnila, Warren B. Moors
本文研究了Wijsman超空间的拓扑性质。研究了Wijsman超空间中孤立点的存在性。证明了每个Tychonoff空间都可以作为闭子空间嵌入局部R的完全度量空间的Wijsman超空间中.
In this paper, topological properties of Wijsman hyperspaces are investigated. We study the existence of isolated points in Wijsman hyperspaces. We show that every Tychonoff space can be embedded as a closed subspace in the Wijsman hyperspace of a complete metric space which is locally R.
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Hereditarily Normal Wijsman Hyperspaces Are Metrizable
本文研究了Wijsman超空间的正规性和度量化。证明了每个遗传正规的Wijsman超空间都是可度量化的。这为1998年Di Maio和Meccariello的问题提供了部分答案。
Wijsman hyperspaces of non-separable metric spaces
给定一个度量空间$\langle X,ρ\rangle$，考虑它的闭集超空间$CL(X)$具有Wijsman拓扑$τ_{W(ρ)}$。已知$\langle{CL(X),τ_{W(ρ)}}\rangle$是可度量化的当且仅当$X$是可分的，并且这是否等价于$\langle{CL(X),τ_{W(ρ)}}\rangle$是正规的是Di Maio和Meccariello的一个公开问题。本文证明了：如果$X$的权是正则不可数基数，且$X$是局部可分的，则$\langle{CL(X),τ_{W(ρ)}}\rangle$不是正规的.我们还解决了Cao，Junnilla和Moors关于Wijsman超空间中孤立点的一些问题。
Since it first emerged in Wijsman's seminal work [29], the Wijsman topology has been intensively studied in the past 50 years. In particular, topological properties of Wijsman hyperspaces, relationships between the Wijsman topology and other hyperspace topologies, and applications of the Wijsman topology in analysis have been explored. However, there are still several fundamental open problems on this topology. In this article, the author gives a brief survey on these problems and some up-to-date partial solutions.
On normality of the Wijsman topology
设$(X,ρ)$是一个度量空间，$(CL(X),W_ρ)$是$X$的所有非空闭子集的超空间，且该超空间具有Wijsman拓扑. Wijsman拓扑是最重要的经典超空间拓扑之一。我们给出了一个问题的部分答案\cite{maio}是否正常的$(CL(X),W_ρ)$是等价于它的度量化。如果$(X,ρ)$是线性度量空间，则$(CL(X),W_ρ)$是正规的当且仅当$(CL(X),W_ρ)$是可度量化的。进一步证明了$CL(X)$上Wijsman拓扑的正规性.
It is shown that if a $T_2$ topological space $X$ contains a closed uncountable discrete subspace, then the spaces $(ω_1 + 1)^ω$ and $(ω_1 + 1)^{ω_1}$ embed into $(CL(X),τ_F)$, the hyperspace of nonempty closed subsets of $X$ equipped with the Fell topology. If $(X,d)$ is a non-separable perfect topological space, then $(ω_1 + 1)^ω$ and $(ω_1 + 1)^{ω_1}$ embed into $(CL(X),τ_{w(d)})$, the hyperspace of nonempty closed subsets of $X$ equipped with the Wijsman topology, giving a partial answer to the Question 3.4 in [CJ].
Let f be a continuous map of a complete separable metric space E onto the irrationals. We show that if a complete separable metric space M contains isometric copies of every closed relatively discrete set in E, then M contains also an isometric copy of some fiber of f. We shall show also that if all fibers of f have positive dimension, then the collection of closed zero-dimensional sets in E is non-analytic in the Wijsman hyperspace of E.
Hyperspace of convex compacta of nonmetrizable compact convex subspaces of locally convex spaces
我们的主要结果是：局部凸空间中紧凸子集$X$的凸紧子集超空间是绝对收缩核的当且仅当$X$是权绝对收缩核的$\leω_1$ .证明了Tychonov立方$I^{ω_1}$的凸紧子集超空间与$I^{ω_1}$同胚.对于$I^{ω_1}$上的锥也证明了类似的结果.我们的证明是基于对紧凸子集超空间映射的分析，特别是证明了这类映射的选择定理。
Hausdorff hyperspaces of $R^m$ and their dense subspaces
设$CLB_H(X)$表示度量空间$X$的闭有界子集的超空间，赋以Hausdorff度量拓扑.我们证明，除其他外，自然稠密子空间的$CLB_H(R^m)$的所有无处稠密闭集，所有完美集，所有康托集和所有勒贝格测度零集是同胚的希尔伯特空间$\ell_2$。此外，我们还研究了Hausdorff度量下实直线$R$的所有非空闭子集的超空间$CL_H(R)$。我们证明了$CL_H(R)$的不可分分量是Hilbert空间$\ell_2(2^{\aleph_0})$同胚的，只要它不包含任何集合$R, [0,\infty), (-\infty,0]$。
We study the relations between a generalization of pseudocompactness, named $(κ, M)$-pseudocompactness, the countably compactness of subspaces of $βω$ and the pseudocompactness of their hyperspaces. We show, by assuming the existence of $\mathfrak c$-many selective ultrafilters, that there exists a subspace of $βω$ that is $(κ, ω^*)$-pseudocompact for all $κ<\mathfrak c$, but $\text{CL}(X)$ isn't pseudocompact. We prove in ZFC that if $ω\subseteq X\subseteq βω$ is such that $X$ is $(\mathfrak c, ω^*)$-pseudocompact, then $\text{CL}(X)$ is pseudocompact, and we further explore this relation by …
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.
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…
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…
Modulo arithmetic of function spaces: Subset hyperspaces as quotients of function spaces
本文研究拓扑空间的子集超空间结构，将特定超空间刻画为函数空间的显式商，讨论相关紧子集超空间的可度量化，证明含Vietoris拓扑的超空间拓扑是含逐点收敛拓扑的函数空间拓扑的商。
An embedding in the Fell topology
证明了若$T_2$拓扑空间包含不可数闭离散集，则$ω_1 \times (ω_1 + 1)$嵌入为$(CL(X),τ_F)$的闭子空间，即$X$的非空闭子集的超空间，该超空间具有Fell拓扑.
The Kantorovich-Rubinshtein metric is an $L^1$-like metric on spaces of probability distributions that enjoys several serendipitous properties. It is complete separable if the underlying metric space of points is complete separable, and in that case it metrizes the weak topology. We introduce a variant of that construction in the realm of quasi-metric spaces, and prove that it is algebraic Yoneda-complete as soon as the underlying quasi-metric space of points is algebraic Yoneda-complete, and that the associated topology is the weak topology. We do this not only for probability distributions, …
On Tychonoff-type hypertopologies
1975年，M. M. Choban在拓扑空间的所有闭子集的集合上引入了一个新的拓扑，它类似于Tychonoff拓扑，但比它弱。Dimov和D. Vakarelov使用了这个新拓扑的推广版本，称之为吉洪诺夫型拓扑。本文研究了集合X的任意子集族M上的Tychonoff型拓扑。当M包含所有单例时，给出了M上所有Tychonoff型拓扑O的描述。用同构定理描述了（M，O）型空间之间特殊形式的连续映射。研究了超空间和子空间关于Tychonoff型拓扑的可积性问题。讨论了具有Tychonoff型拓扑O的超空间（M，O）的一些拓扑性质。
A Universal Quasi-Polish Space
本文构造了通用拟波兰空间Y，证明所有拟波兰空间均同胚于Y的闭子空间，其超空间F(Y)为拟波兰空间，且Y的波兰闭子空间集合是F(Y)的Π¹₁完全余解析非博雷尔子集。
A regular topological space $X$ is defined to be a $\mathfrak P_0$-space if it has countable Pytkeev network. A network $\mathcal N$ for $X$ is called a Pytkeev network if for any point $x\in X$, neighborhood $O_x\subset X$ of $x$ and subset $A\subset X$ accumulating at a $x$ there is a set $N\in\mathcal N$ such that $N\subset O_x$ and $N\cap A$ is infinite. The class of $\mathfrak P_0$-spaces contains all metrizable separable spaces and is (properly) contained in the Michael's class of $\aleph_0$-spaces. It is closed under many topological operations: taking subspaces, countable Tychonoff pro…
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