Many classically used function space structures (including the topology of pointwise convergence, the compact-open topology, the Isbell topology and the continuous convergence) are induced by a hyperspace structure counterpart. This scheme is used to study local properties of function space structures on $C(X,\mathbb R)$, such as character, tighntess, fan-tightness, strong fan-tightness, the Fr{é}chet property and some of its variants. Under mild conditions, local properties of $C(X,\mathbb R)$ at the zero function correspond to the same property of the associated hyperspace structure at $X$. The latter is often easy to characterize in terms of covering properties of $X$. This way, many classical results are recovered or refined, and new results are obtained. In particular, it is shown that tightness and character coincide for the continuous convergence on $C(X,\mathbb R)$ and is equal to the Lindel{ö}f degree of $X$. As a consequence, if $X$ is consonant, the tightness of $C(X,\mathbb R)$ for the compact-open topology is equal to the Lindel{ö}f degree of $X$.
Our main result states that the hyperspace of convex compact subsets of a compact convex subset $X$ in a locally convex space is an absolute retract if and only if $X$ is an absolute retract of weight $\leω_1$. It is also proved that the hyperspace of convex compact subsets of the Tychonov cube $I^{ω_1}$ is homeomorphic to $I^{ω_1}$. An analogous result is also proved for the cone over $I^{ω_1}$. Our proofs are based on analysis of maps of hyperspaces of compact convex subsets, in particular, selection theorems for such maps are proved.
We introduce and study the topological concepts of chain transitivity, mixing and chain mixing property for dynamical systems induced by uniform hyperspaces. These notions generalize the relevant concepts for metric spaces.
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.
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.
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.
Let $X$ be a Hausdorff space and let $\mathcal{H}$ be one of the hyperspaces $CL(X)$, $\mathcal{K}(X)$, $\mathcal{F}(X)$ or $\mathcal{F}_n(X)$ ($n$ a positive integer) with the Vietoris topology. We study the following disconnectedness properties for $\mathcal{H}$: extremal disconnectedness, being a $F^\prime$-space, $P$-space or weak $P$-space and hereditary disconnectedness. Our main result states: if $X$ is Hausdorff and $F\subset X$ is a closed subset such that $(a)$ both $F$ and $X-F$ are totally disconnected, $(b)$ the quotient $X/F$ is hereditarily disconnected, then $\mathcal{K}(X)$ is…
In this paper we point out an interesting geometric structure of nonnegative metric curvature emerging from the hyperspaces of decomposable, non-locally connected homogeneous continua, where "smooth" and "non-smooth" partitions live together, similarly to the macroscopic and the quantum realms of the Universe.
In this paper, we shall study categorial properties of the hyperspace of all nontrivial convergent sequences $\mathcal{S}_c(X)$ of a Freéch-Urysohn space $X$, this hyperspace is equipped with the Vietoris topology. We mainly prove that $\mathcal{S}_c(X)$ is meager whenever $X$ is a crowded space, as a corollary we obtain that if $\mathcal{S}_c(X)$ is Baire, the $X$ has a dense subset of isolated points. As an interesting example $\mathcal{S}_c(ω_1)$ has the Baire property, where $ω_1$ carries the order topology (this answers a question from \cite{sal-yas}). We can give more examples like this …
It is shown that the hyperspace of all nonempty closed subsets $\Cld_{AW}(X)$ of a separable metric space $X$ endowed with the Attouch-Wets topology is homeomorphic to a separable Hilbert space if and only if the completion of $X$ is proper, locally connected and contains no bounded connected component, $X$ is topologically complete and not locally compact at infinity.
The inclusion hyperspace functor, the capacity functor and monads for these functors have been extended from the category of compact Hausdorff spaces to the category of Tychonoff spaces. Properties of spaces and maps of inclusion hyperspaces and capacities (non-additive measures) on Tychonoff spaces are investigated.
This paper is a continuation of the work done in \cite{sal-yas} and \cite{may-pat-rob}. We deal with the Vietoris hyperspace of all nontrivial convergent sequences $\mathcal{S}_c(X)$ of a space $X$. We answer some questions in \cite{sal-yas} and generalize several results in \cite{may-pat-rob}. We prove that: The connectedness of $X$ implies the connectedness of $\mathcal{S}_c(X)$; the local connectedness of $X$ is equivalent to the local connectedness of $\mathcal{S}_c(X)$; and the path-wise connectedness of $\mathcal{S}_c(X)$ implies the path-wise connectedness of $X$. We also show that the …
The symbol $\mathcal{S}(X)$ denotes the hyperspace of finite unions of convergent sequences in a Hausdorff space $X$. This hyperspace is endowed with the Vietoris topology. First of all, we give a characterization of convergent sequence in $\mathcal{S}(X)$. Then we consider some cardinal invariants on $\mathcal{S}(X)$, and compare the character, the pseudocharacter, the $sn$-character, the $so$-character, the network weight and $cs$-network weight of $\mathcal{S}(X)$ with the corresponding cardinal function of $X$. Moreover, we consider rank $k$-diagonal on $\mathcal{S}(X)$, and give a space $…
Assume that $X$ is a regular space. We study topological properties of the family $S_c(X)$ of nontrivial convergent sequences in $X$ equipped with the Vietoris topology. We show that if $X$ has no isolated points, then $S_c(X)$ is a space of the first category which answers the question posed by S. Garcia-Ferreira and Y.F. Ortiz-Castillo.
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…
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…
We study the problem of topologically order-embedding a given topological poset X in the space of all closed subsets of X which is topologized by the Fell topology and ordered by set inclusion. We show that this can be achieved whenever X is a topological semilattice (resp. lattice) or a topological po-group, and X is locally compact and order-connected (resp. connected). We give limiting examples to show that these results are tight, and provide several applications of them. In particular, a locally compact version of the Urysohn-Carruth metrization theorem is obtained, a new fixed point theo…
Given a metric space $\langle X,ρ\rangle$, consider its hyperspace of closed sets $CL(X)$ with the Wijsman topology $τ_{W(ρ)}$. It is known that $\langle{CL(X),τ_{W(ρ)}}\rangle$ is metrizable if and only if $X$ is separable and it is an open question by Di Maio and Meccariello whether this is equivalent to $\langle{CL(X),τ_{W(ρ)}}\rangle$ being normal. In this paper we prove that if the weight of $X$ is a regular uncountable cardinal and $X$ is locally separable, then $\langle{CL(X),τ_{W(ρ)}}\rangle$ is not normal. We also solve some questions by Cao, Junnilla and Moors regarding isolated poin…
For a regular space $X$, the hyperspace $(CL(X), τ_{F})$ (resp., $(CL(X), τ_{V})$) is the space of all nonempty closed subsets of $X$ with the Fell topology (resp., Vietoris topology). In this paper, we give the characterization of the space $X$ such that the hyperspace $(CL(X), τ_{F})$ (resp., $(CL(X), τ_{V})$) with a countable character of closed subsets. We mainly prove that $(CL(X), τ_F)$ has a countable character on each closed subset if and only if $X$ is compact metrizable, and $(CL(X), τ_F)$ has a countable character on each compact subset if and only if $X$ is locally compact and sepa…
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.
The hyperspace of all nontrivial convergent sequences in a Hausdorff space $X$ is denoted by $\mathcal{S}_c(X)$. This hyperspace is endowed with the Vietoris topology. In connection with a question and a problem by García-Ferreira, Ortiz-Castillo and Rojas-Hernández, concerning conditions under which $\mathcal S_c(X)$ is pathwise connected, in the current paper we study the latter property and the contractibility of $\mathcal{S}_c(X)$. We present necessary conditions on a space $X$ to obtain the path connectedness of $\mathcal{S}_c(X)$. We also provide some sufficient conditions on a space $X$…
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, …
For a given compact Hausdorff space $X$, we construct the space $OS_{f}(X)$ of normed, order-preserving, weakly additive, positively homogeneous and semi-additive functionals (for brevity, semi-additive functionals) and it is proved that the hyperspace $\exp\, X$ of the space $X$ is a deformation retract of the constructed space. Further we show that the shapes of the spaces $OS_{f}(X)$ and $\exp\, X$ coincide. We establish that if the space $\exp\, X$ is contractible, then the space $OS_{f}(X)$ is also contractible.
In 1975, M. M. Choban introduced a new topology on the set of all closed subsets of a topological space, similar to the Tychonoff topology but weaker than it. In 1998, G. Dimov and D. Vakarelov used a generalized version of this new topology, calling it Tychonoff-type topology. The present paper is devoted to a detailed study of Tychonoff-type topologies on an arbitrary family M of subsets of a set X. When M contains all singletons, a description of all Tychonoff-type topologies O on M is given. The continuous maps of a special form between spaces of the type (M,O) are described in an isomorph…
This paper is a sequel of Imamura (2019) (arXiv:1711.01609) where we set up a framework of nonstandard large-scale topology. In the present paper, we apply our framework to various topics in large-scale topology: spaces having with both small-scale and large-scale structures, large-scale structures on nonstandard extensions, size properties of subsets of coarse spaces, and coarse hyperspaces.
Never has an issue of the SPM Bulletin contained as much interesting information as this issue does. In addition to the interesting research announcements, this issue contains announcements of solutions for three open problems, one of which being a Problem of the Month in an earlier issue. CONTENTS: The Reznichenko property and the Pytkeev property in hyperspaces; Between Maharam's and von Neumann's problems; Fraisse Limits, Ramsey Theory, and Topological Dynamics of Automorphism Groups; Concerning problems about cardinal invariants on Boolean algebras; Small Locally Compact Linearly Lindelof …
We give conditions under which a product of topological spaces satisfies some local property. The conditions are necessary and sufficient when the corresponding global property is preserved under finite products. Further examples include local sequential compactness, local Lindelöfness, the local Menger property.
An abstract scheme using particular types of relations on filters leads to general unifying results on stability under supremum and product of local topological properties. We present applications for Frechetness, strong Frechetness, countable tightness and countable fan-tightness, some of which recover or refine classical results, some of which are new. The reader may find other applications as well.
We investigate, and prove equivalent, effective versions of local connectivity and uniformly local arcwise connectivity for connected and computably compact subspaces of Euclidean space. We also prove that Euclidean continua that are computably compact and effectively locally connected are computably arcwise connected.
We prove that for a compact subgroup $H$ of a locally compact Hausdorff group $G$, the following properties are mutually equivalent: (1) $G/H$ is a manifold, (2) $G/H$ is finite-dimensional and locally connected, (3) $G/H$ is locally contractible, (4) $G/H$ is an ANE for paracompact spaces, (5) $G/H$ is a metrizable $G$-ANE for paracompact proper $G$-spaces having a paracompact orbit space. A new version of the Approximate slice theorem is also proven in the light of these results.
