J. F. Peters, C. Guadagni
This article introduces strongly near proximity, which represents a new kind of proximity called \emph{almost proximity}. A main result in this paper is the introduction of a hit-and-miss topology on ${CL}(X)$, the hyperspace of nonempty closed subsets of $X$, based on the strongly near proximity.
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This article introduces strongly far proximity, which is associated with Lodato proximity $δ$. A main result in this paper is the introduction of a hit-and-miss topology on $\mbox{CL}(X)$, the hyperspace of nonempty closed subsets of $X$, based on the strongly far proximity.
We prove that the inclusion of map(X,Y) into map(K(X),K(Y)) is continuous, where K(X) is the space of non-empty compact subsets of X (also known as the hyperspace of compact subsets of X), and both spaces of maps are endowed with the compact-open topology.
In this paper, we prove a theorem about embedding of some partially ordered topological spaces in topological hyperspaces equipped with Fell topology. Then we give some examples to show that the map defining the embedding may not be continuous with respect to Vietoris topology or Hausdorff topology equipped on the hyperspaces.
We study the heredity of the classes of generalized metric spaces (for example, spaces with a $σ$-hereditarily closure-preserving $k$-network, spaces with a point-countable base, spaces with a base of countable order, spaces with a point-regular base, Nagata-spaces, $c$-semi-stratifiable spaces, $γ$-spaces, semi-metrizable spaces) to the hyperspaces of nonempty compact subsets and finite subsets with the Vietoris topology.
We introduce a new Vietoris-type hypertopology by means of the upper-Vietoris-type hypertopology defined by G. Dimov and D. Vakarelov [On Scott consequence systems, Fundamenta Informaticae, 33 (1998), 43-70] (it was called there {\em Tychonoff-type hypertopology}) and the lower-Vietoris-type hypertopology introduced by E. Ivanova-Dimova [Lower-Vietoris-type topologies on hyperspaces, Topology Appl. (to appear)]. We study this new Vietoris-type hypertopology and show that it is, in general, different from the Vietoris topology. Also, some of the results of E. Michael [Topologies on spaces of su…
For a topological space $X$, let $CL(X)$ be the set of all non-empty closed subset of $X$, and denote the set $CL(X)$ with the Vietoris topology by $(CL(X), \mathbb{V})$. In this paper, we mainly discuss the hyperspace $(CL(X), \mathbb{V})$ when $X$ is an infinite countable discrete space. As an application, we first prove that the hyperspace with the Vietoris topology on an infinite countable discrete space contains a closed copy of $n$-th power of Sorgenfrey line for each $n\in\mathbb{N}$. Then we investigate the tightness of the hyperspace $(CL(X), \mathbb{V})$, and prove that the tightness…
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。
For a Hausdorff space $X$ we denote be $2^X$ the family of all closed subsets of $X$. In this paper we continue to research relationships between closure -type properties of hyperspaces over a space $X$ and covering properties of $X$. We investigate selectors for sequence of subsets of the space $2^{X}$ with the $Z^{+}$-topology and the upper Fell topology.
When one considers the collection $\mathcal{H}(\mathbb{R}^n)$ of all compact subsets of $\mathbb{R}^n$ and equip it with a topology, many questions can be asked about the topological space one ends up with. This is an example of a hyperspace, a mathematical object which has been studied in a more abstract setting since the beginning of the 20th century. Here we give an elementary proof of the path-connectedness of $\mathcal{H}(\mathbb{R}^n)$, with the topology induced by the Hausdoff metric, by exploring the vector structure of $\mathbb{R}^n$ and using only basic ideas of topology of metric sp…
Hyperspaces with the Attouch-Wets topology homeomorphic to $l_2$
证明了赋Attouch-Wets拓扑的可分度量空间$X$的所有非空闭子集$\Cld_{AW}(X)$的超空间与可分希尔伯特空间同胚当且仅当$X$的完备化是真的、局部连通的且不含有界连通分支，$X$是拓扑完备的且在无穷远处不是局部紧的.
We introduce a new lower-Vietoris-type hypertopology in a way similar to that with which a new upper-Vietoris-type hypertopology was introduced in G. Dimov and D. Vakarelov, "On Scott consequence systems", Fundamenta Informaticae, 33 (1998), 43-70. (it was called there {\em Tychonoff-type hypertopology}). We study this new hypertopology and, in particular, we generalize many results from E. Cuchillo-Ibanez, M. A. Moron and F. R. Ruiz del Portal, "Lower semifinite topology in hyperspaces", Topology Proceedings, 17 (1992), 29-39. As a corollary, we get that for every continuous map $f:X\longrigh…
Since it first emerged in Wijsman's seminal work [29], the Wijsman topology has been intensively studied in the past 50 years. In particular, topological properties of Wijsman hyperspaces, relationships between the Wijsman topology and other hyperspace topologies, and applications of the Wijsman topology in analysis have been explored. However, there are still several fundamental open problems on this topology. In this article, the author gives a brief survey on these problems and some up-to-date partial solutions.
Hyperspaces of Closed Limit Sets
研究了拓扑空间的闭极限子集集上的Michael拓扑和Fell拓扑。特别注意的是所有极大极限集的子族。
The topology of systems of hyperspaces determined by dimension functions
给定一个非退化Peano连续统$X$，一个定义在$X$的紧子集族$2^X_*$上的维数函数$D:2^X_*\to[0,\infty]$，以及一个子集$Γ\subset[0,\infty)$，我们认识到系统$(2^X,\D_{\leγ}(X))_{α\inΓ}$的拓扑结构，其中$2^X$是$X$的非空紧子集的超空间，$D_{\leγ}(X)$是$2^X$的子空间，由非空紧子集$K\subset X$和$D(K)\leγ$组成.
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 $X$.
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.
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, …
We study Vietoris hyperspaces of closed and closed final sets of Priestley spaces. We are particularly interested in Skula topologies. A topological space is \emph{Skula} if its topology is generated by differences of open sets of another topology. A compact Skula space is scattered and moreover has a natural well-founded ordering compatible with the topology, namely, it is a Priestley space. One of our main objectives is investigating Vietoris hyperspaces of general Priestley spaces, addressing the question when their topologies are Skula and computing the associated ordinal ranks. We apply o…
The quasi-metrizability of hyperspaces
本文研究了赋予不同拓扑的超空间的拟可度量性，分别给出了 Vietoris、局部有限和 Fell 拓扑下超空间拟可度量或可对称化的充要条件，并据此对相关猜想给出了否定回答。
A unified theory of function spaces and hyperspaces: local properties
许多经典的函数空间结构（包括点态收敛的点式收敛拓扑、紧开拓扑、Isbell拓扑和连续收敛拓扑）都是由对应的超空间结构导出的。利用该方案研究了$C(X,\mathbb R)$上函数空间结构的局部性质，如特征、紧性、扇紧性、强扇紧性、Fr{é}chet性质及其一些变体.在温和的条件下，$C(X,\mathbb R)$在零函数处的局部性质对应于相关超空间结构在$X$处的相同性质。后者通常容易根据$X$的覆盖性质来表征。这样，许多经典的结果被恢复或改进，并获得新的结果。特别地，证明了$C(X,\mathbb R)$上连续收敛的紧度和特征一致，并且与$X$的Lindel{<$}f度相等.因此，如果$X$是辅音的，那么$C(X,\mathbb R)$对于紧开拓扑的紧度等于$X$的Lindel{<$}f度。
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。
This article introduces strongly proximal continuous (s.p.c.) functions, strong proximal equivalence (s.p.e.) and strong connectedness. A main result is that if topological spaces $X,Y$ are endowed with compatible strong proximities and $f:X\longrightarrow Y$ is a bijective s.p.e., then its extension on the hyperspaces $\CL(X)$ and $\CL(Y)$, endowed with the related strongly hit and miss hypertopologies, is a homeomorphism. For a topological space endowed with a strongly near proximity, strongly proximal connectedness implies connectedness but not conversely. Conditions required for strongly p…
We consider three monads on Top, the category of topological spaces, which formalize topological aspects of probability and possibility in categorical terms. The first one is the Hoare hyperspace monad H, which assigns to every space its space of closed subsets equipped with the lower Vietoris topology. The second is the monad V of continuous valuations, also known as the extended probabilistic powerdomain. We construct both monads in a unified way in terms of double dualization. This reveals a close analogy between them, and allows us to prove that the operation of taking the support of a con…
This article introduces strongly near smooth manifolds. The main results are (i) second countability of the strongly hit and far-miss topology on a family $\mathcal{B}$ of subsets on the Lodato proximity space of regular open sets to which singletons are added, (ii) manifold strong proximity, (iii) strong proximity of charts in manifold atlases implies that the charts have nonempty intersection. The application of these results is given in terms of the nearness of atlases and charts of proximal manifolds and what are known as Vorono\" i manifolds.
In this paper, we introduce the notion of topologically Banach contraction mapping defined on an arbitrary topological space X with the help of a continuous function $g:X\times X\rightarrow \mathbb{R}$ and investigate the existence of fixed points of such mapping. Moreover, we introduce two types of mappings defined on a non-empty subset of X and produce sufficient conditions which will ensure the existence of best proximity points for these mappings. Our best proximity point results also extend some existing results from metric spaces or Banach spaces to topological spaces. More precisely, ou…
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