Katsuhisa Koshino
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.
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The Reznichenko property and the Pytkeev property in hyperspaces
本文研究了超空间拓扑中的两个闭包性质：Reznichenko性质和Pytkeev性质。
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.
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。
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 $X$.
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。
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…
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.
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…
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.
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…
Calibers of canonical hyperconnected spaces and their Vietoris hyperspaces
本文计算了特定超连通空间的尺度，并证明其与Vietoris超空间的尺度集合一致，为超空间理论提供了新结果。
The Hurewicz Property and the Vietoris Hyperspace
本文利用 $k$-覆盖与可成群性刻画 Vietoris 超空间的 Hurewicz 性质，并给出有限幂及连续函数空间的相关等价条件，回答 Kočinac 的问题。
Recently, David S. Lipham has shown that if $X$ is an Erdős space factor then the Vietoris hyperspace $\mathcal{F}(X)$ of finite subsets of $X$ is an Erdős space factor. In this short note we prove that if $\mathfrak{E}$ denotes Erdős space then $\mathcal{F}(\mathfrak{E})$ is in fact homeomorphic to $\mathfrak{E}$.
In this paper we consider the hyperspace $C_{n}(X)$ of non-empty and closed subsets of a base space $X$ with up to $n$ connected components. We consider a class of base spaces called finite ray-graphs, which are a noncompact variation on finite graphs. We prove two results about the structure of these hyperspaces under different topologies (Hausdorff and Vietoris), in particular, about their number of connected components.
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 $D(K)\leγ$.
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…
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$R, [0,\infty), (-\infty,0]$。
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.
The notion of max-plus convex subset of Euclidean space can be naturally extended to other linear spaces. The aim of this paper is to describe the topology of hyperspaces of max-plus convex subsets of Tychonov powers $\mathbb R^τ$ of the real line. We show that the corresponding spaces are AR's if and only if $τ\leω_1$.
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…
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.
We say that two classes of topological spaces are equivalent if each member of one class has a homeomorphic copy in the other class and vice versa. Usually when the Borel complexity of a class of metrizable compacta is considered, the class is realized as the subset of the hyperspace $\mathcal{K}([0, 1]^ω)$ containing all homeomorphic copies of members of the given class. We are rather interested in the lowest possible complexity among all equivalent realizations of the given class in the hyperspace. We recall that to every analytic subset of $\mathcal{K}([0, 1]^ω)$ there exist an equivalent $…
On the Variety of Hyperspace Selections
本文研究Vietoris超空间连续选择点的集合$\mathcal{O}_{cs}(X)$的性质，证明其闭包是完全不连通子集，并在第一可数完全不连通空间中该集合是闭集。
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…
Given a continuum $X$, let $\mathcal{NB} (\mathcal{F}_1(X))$ be the hyperspace of nonblockers of $\mathcal{F}_1(X)$. In this paper, we show that if $X$ is hereditarily decomposable with the property of Kelley such that $\mathcal{NB} (\mathcal{F}_1(X))$ is a continuum, then $X$ is a simple closed curve. Thus, we characterize the simple closed curve as the unique hereditarily decomposable continuum with the property of Kelley $X$ such that its hyperspace $\mathcal{NB} (\mathcal{F}_1(X))$ is a continuum.
Characterizing metric spaces whose hyperspaces are absolute neighborhood retracts
我们刻画了度量空间$X$，它的超空间$2^X$或$Bd(X)$的非空闭（有界）子集，赋予Hausdorff度量，是绝对[邻域]收缩.
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 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.
An Excursion with Divergence Properties
本文比较了各种选择发散性质，如离散选择性和选择性高发散性，并研究了有限子集超空间的某些影响，以及如何通过特定的选择游戏来表征闭合离散选择游戏。
Butterfly Points and Hyperspace Selections
本文通过局部性质刻画了Vietoris超空间的连续选择在全集上的取值点，证明其为强蝴蝶点当且仅当存在可数闭开基，并完全解决了作者先前的问题，同时给出了缺乏该性质时的更强选择。
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 …
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…
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…
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…
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.
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].
Plasticity in graph metric spaces and their hyperspaces
本文证明连通的局部有限正则图及其与紧度量空间的乘积的超空间在豪斯多夫度量下是可塑的，并给出相关判据与反例。
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。
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。
For a given continuum $X$ and a natural number $n,$ we consider the hyperspace $F_n(X)$ of all nonempty subsets of $X$ with at most $n$ points, metrized by the Hausdorff metric. In this paper we show that if $X$ is a dendrite whose set of end points is closed, $n \in \mathbb{N}$ and $Y$ is a continuum such that the hyperspaces $F_n(X)$ and $F_n(Y)$ are homeomorphic, then $Y$ is a dendrite whose set of end points is closed.
Let $(X,ρ)$ be a metric space and $(CL(X),W_ρ)$ be the hyperspace of all nonempty closed subsets of $X$ equipped with the Wijsman topology. The Wijsman topology is one of the most important classical hyperspace topologies. We give a partial answer to a question posed in \cite{maio} whether the normality of $(CL(X),W_ρ)$ is equivalent to its metrizability. If $(X,ρ)$ is a linear metric space, then $(CL(X),W_ρ)$ is normal if and only if $(CL(X),W_ρ)$ is metrizable. Some further results concerning normality of the Wijsman topology on $CL(X)$ are also proved.
Spectre operator, achievement sets and sets of P-sums in a hyperspace of compact sets
本文研究了在紧集超空间中，基于谱定义的算子S及其相关成就集和P-和集的性质，探讨了其在平面中的特性。
The characterizations of hyperspaces and free topological groups with an $ω^ω$-base
本文刻画了自由阿贝尔拓扑群及超空间具有 $\omega^\omega$-基的条件，给出了关于 $k$-空间、可度量空间和 Polish 空间的等价刻画。
Mean dimension explosion of induced homeomorphisms
本文研究了诱导超空间映射的平均维数爆炸现象，发现基础系统拓扑熵为零时诱导映射平均维数可为无穷大，并分别给出了Morse-Smale微分同胚、圆周同胚及度量平均维数的相关条件与结论。
Hyperspace convergences, bornologies and geometric set functionals
本文研究了度量空间超空间中由距离泛函一致收敛诱导的$τ_{\mathcal{S},d}$-收敛与有界收敛、弱收敛之间的关系，利用正Lipschitz函数对集合逐点放大的思想作为主要工具，并为Attouch-Wets收敛的已知结果提供了新证明。
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。
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 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 $X$…
We explore the connections between selection games on Hausdorff spaces and their corresponding Vietoris space of compact subsets. These considerations offer a similar relationship as the well-known relationship between $ω$-covers of $X$ and regular open covers of the finite powers of $X$. The primary utility of this method is to establish similar relationships with $k$-covers and the Vietoris space of compact subsets. Particularly, we show that some commonly studied selection principles are equivalent to a related hyperspace being Menger or Rothberger. We then apply these equivalences to corre…
In 2017, Tkachuk isolated the closed discrete selection property while working on problems related to function spaces [15]. In this paper we will study the closed discrete selection property and the related games and strategies on $C_k(X)$. Clontz and Holshouser showed previously that the closed discrete selection game on $C_p(X)$ is equivalent to a modification of the point-open game on $X$. In this paper we show that the closed discrete selection game on $C_k(X)$ is equivalent to a modification of the compact-open game on $X$. We also connect discrete selection properties on $C_k(X)$ to a va…
Characterizations of Urysohn universal ultrametric spaces
本研究通过闭球无穷等距子集刻画超度量空间的有限内射性与Urysohn万有超度量空间，证明笛卡尔积、超空间、对称积保持该空间结构，且完备有限内射超度量空间必含其等距子空间。
Set convergences and uniform convergence of distance functionals on a bornology
本文基于 bornology 研究度量空间闭子集超空间在距离泛函一致收敛拓扑下的特征，并引入新 bornology 类与度量拓扑。
A topological space $X$ has the strong Pytkeev property at a point $x\in X$ if there exists a countable family $\mathcal N$ of subsets of $X$ such that for each neighborhood $O_x\subset X$ and subset $A\subset X$ accumulating at $x$, there is a set $N\in\mathcal N$ such that $N\subset O_x$ and $N\cap A$ is infinite. We prove that for any $\aleph_0$-space $X$ and any space $Y$ with the strong Pytkeev property at a point $y\in Y$ the function space $C_k(X,Y)$ has the strong Pytkeev property at the constant function $X\to \{y\}\subset Y$. If the space $Y$ is rectifiable, then the function space $…
Characterizing function spaces which have the property (B) of Banakh
本文刻画了具有Banakh性质(B)的函数空间，证明了$C_p(X)$不满足性质(B)当且仅当空间$X$具有性质$(\kappa)$，并对紧开拓扑给出类似刻画，回答了Tkachuk的问题。
Borel complexity of the family of attractors for weak IFSs
本文研究弱迭代函数系统吸引子族的 Borel 复杂度，证明 wIFS$^d$ 是 $G_{δσ}$-难且非 $F_{σδ}$，并在一维情形下证明其为解析子集，从而刻画了其与经典 IFS 吸引子族的复杂度差异。
All spaces are assumed to be separable and metrizable. Our main result is that the statement "For every space $X$, every closed subset of $X$ has the perfect set property if and only if every analytic subset of $X$ has the perfect set property" is equivalent to $\mathfrak{b}>ω_1$ (hence, in particular, it is independent of $\mathsf{ZFC}$). This, together with a theorem of Solecki and an example of Miller, will allow us to determine the status of the statement "For every space $X$, if every $\mathbfΓ$ subset of $X$ has the perfect set property then every $\mathbfΓ'$ subset of $X$ has the perfec…
We show that even for subsets X of the real line which do not contain perfect sets, the Hurewicz property does not imply the property S1(Gamma,Gamma), asserting that for each countable family of open gamma-covers of X, there is a choice function whose image is a gamma-cover of X. This settles a problem of Just, Miller, Scheepers, and Szeptycki. Our main result also answers a question of Bartoszynski and Tsaban, and implies that for C_p(X), the conjunction of Sakai's strong countable fan tightness and the Reznichenko property does not imply Arhangelskii's property alpha_2.
We discuss a set-valued generalization of strong proximinality in Banach spaces, introduced by J. Mach [Continuity properties of Chebyshev centers, J. Approx. Theory, 29(3):223--230, 1980] as property-$(P_1)$. We establish that if the closed unit ball of a closed subspace of a Banach space $X$ possesses property-$(P_1)$ for each of the classes of closed bounded, compact and finite subsets of $X$, then so does the subspace. It is also proved that the closed unit ball of an $M$-ideal in an $L_{1}$-predual space satisfies property-$(P_{1})$ for the compact subsets of the space. For a Choquet simp…
We extend the well-known Gelfand-Phillips property for Banach spaces to locally convex spaces, defining a locally convex space $E$ to be Gelfand-Phillips if every limited set in $E$ is precompact in the topology on $E$ defined by barrels. Several characterizations of Gelfand-Phillips spaces are given. The problem of preservation of the Gelfand-Phillips property by standard operations over locally convex spaces is considered. Also we explore the Gelfand-Phillips property in spaces $C(X)$ of continuous functions on a Tychonoff space $X$. If $τ$ and $\mathcal T$ are two locally convex topologies …
We show that every Abelian group satisfying a mild cardinal inequality admits a pseudocompact group topology from which all countable subgroups inherit the maximal totally bounded topology (we say that such a topology satisfies property $\h$). Every pseudocompact Abelian group $G$ with cardinality $|G|\leq 2^{2^\cc}$ satisfies this inequality and therefore admits a pseudocompact group topology with property $\h$. Under the Singular Cardinal Hypothesis (SCH) this criterion can be combined with an analysis of the algebraic structure of pseudocompact groups to prove that every pseudocompact Abeli…
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