Sagarmoy Bag, Ram Chandra Manna, Sourav Kanti Patra
We have defined almost separable space. We show that like separability, almost separability is $c$ productive and converse also true under some restrictions. We establish a Baire Category theorem like result in Hausdorff, Pseudocompacts spaces. We investigate few relationships among separability, almost separability, sequential separability, strongly sequential separability.
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A subspace Y of a separable metrizable space X is separable, but without X metrizable this is not true even If Y is a closed linear subspace of a topological vector space X. K.H. Hofmann and S.A. Morris introduced the class of pro-Lie groups which consists of projective limits of finite-dimensional Lie groups and proved that it contains all compact groups, locally compact abelian groups and connected locally compact groups and is closed under products and closed subgroups. A topological group G is almost connected if the quotient group of G by the connected component of its identity is compact…
We investigate a relations of almost isometric embedding and almost isometry between metric spaces and prove that with respect to these relations: (1) There is a countable universal metric space. (2) There may exist fewer than continuum separable metric spaces on aleph_1 so that every separable metric space is almost isometrically embedded into one of them when the continuum hypothesis fails. (3) There is no collection of fewer than continuum metric spaces of cardinality aleph_2 so that every ultra-metric space of cardinality aleph_2 is almost isometrically embedded into one of them if aleph_2…
We define scoring metrics on separable metric spaces and show that they are always no coarser than the metrics from which they spring.
A space has $σ$-compact tightness if the closures of $σ$-compact subsets determines the topology. We consider a dense set variant that we call densely k-separable. We consider the question of whether every densely k-separable space is separable. The somewhat surprising answer is that this property, for compact spaces, implies that every dense set is separable. The path to this result relies on the known connections established between $π$-weight and the density of all dense subsets, or more precisely, the cardinal invariant $δ(X)$.
We investigate the existence of well-ordered sequences of Baire 1 functions on separable metric spaces.
We explore almost-normality in Isbell-Mrówka spaces and some related concepts. We use forcing to provide an example of an almost-normal not normal almost disjoint family, explore the concept of semi-normality in Isbell-Mrówka spaces, define the concept of strongly $(\aleph_0, <\mathfrak c)$-separated almost disjoint families and prove the generic existence of completely separable strongly $(\aleph_0, <\mathfrak c)$-almost disjoint families assuming $\mathfrak s=\mathfrak c$ and $\mathfrak b=\mathfrak c$. We also provide an example of a Tychonoff almost-normal not normal pseudocompact space whi…
Denote by Q(k) a σ-discrete metric weight-homogeneous space of weight k. We give an internal description of the space Q(k)^ω. We prove that the Baire space B(k) is densely homogeneous with respect to Q(k)^ωif k > ω. Properties of some non-separable h-homogeneous absolute F_{σδ}-sets and G_{δσ}-sets are investigated.
A space $X$ is called selectively separable(R-separable) if for every sequence of dense subspaces $(D_n : n\inω)$ one can pick finite (respectively, one-point) subsets $F_n\subset D_n$ such that $\bigcup_{n\inω}F_n$ is dense in $X$. These properties are much stronger than separability, but are equivalent to it in the presence of certain convergence properties. For example, we show that every Hausdorff separable radial space is R-separable and note that neither separable sequential nor separable Whyburn spaces have to be selectively separable. A space is called \emph{d-separable} if it has a de…
New examples in the study of selectively separable spaces
本文针对可数正则 wH-separable 但非 H-separable 的空间，将已有 Martin 公理下的结果改进为 ZFC 例子，并在更弱假设下给出两个兼具相关性质的新构造。
The notions of a proper dyadic subbase and an independent subbase was introduced by H. Tsuiki to investigate in {0, 1, bot}-sequence codings of topological spaces. We show that every separable metrizable space has a proper dyadic subbase whose restriction to the perfect set defined by the Cantor-Bendixson theorem forms an independent subbase of the restricted space.
In this paper, we characterize infinite-dimensional manifolds modeled on absorbing sets in non-separable Hilbert spaces by using the discrete cells property, which is a general position property. Moreover, we study the discrete (locally finite) approximation property, which is an extension of the discrete cells property.
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.
Labeled Trees Generating Separable and Locally Finite Ultrametrics
本文研究标记树与其生成的超度量空间之间的关系，刻画了生成可分及局部有限超度量空间的标记树，并建立了局部有限情形下 König 无穷引理的类比。
A topological space $X$ is said to be {\em $Y$-rigid} if any continuous map $f:X\rightarrow Y$ is constant. In this paper we construct a number of examples of regular countably compact $\mathbb R$-rigid spaces with additional properties like separability and first countability. This way we answer several questions of Tzannes, Banakh, Ravsky, as well as get a consistent example of $\mathbb R$-rigid Nyikos space. Also, we show that it is consistent with ZFC that for every cardinal $κ<\mathfrak c$ there exists a regular separable countably compact space $X$ which is $Y$-rigid with respect to any …
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…
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 $X$.
For every space $X$ let $\mathcal K(X)$ be the set of all compact subsets of $X$. Christensen \cite{c:74} proved that if $X, Y$ are separable metrizable spaces and $F\colon\mathcal{K}(X)\to\mathcal{K}(Y)$ is a monotone map such that any $L\in\mathcal{K}(Y)$ is covered by $F(K)$ for some $K\in\mathcal{K}(X)$, then $Y$ is complete provided $X$ is complete. It is well known \cite{bgp} that this result is not true for non-separable spaces. In this paper we discuss some additional properties of $F$ which guarantee the validity of Christensen's result for more general spaces.
We provide a ZFC example of a compact space K such that C(K)* is w*-separable but its closed unit ball is not w*-separable. All previous examples of such kind had been constructed under CH. We also discuss the measurability of the supremum norm on that C(K) equipped with its weak Baire sigma-algebra.
We prove that in the Miller model, every $M$-separable space of the form $C_p(X)$, where $X$ is metrizable and separable, is productively $M$-separable, i.e., $C_p(X)\times Y$ is $M$-separable for every countable $M$-separable $Y$.
We prove that in the Laver model for the consistency of the Borel's conjecture, the product of any two $H$-separable spaces is $M$-separable.
A separable space is strongly sequentially separable if, for each countable dense set, every point in the space is a limit of a sequence from the dense set. We consider this and related properties, for the spaces of continous and Borel real-valued functions on Tychonoff spaces, with the topology of pointwise convergence. Our results solve a problem stated by Gartside, Lo, and Marsh.
Sequentially compact separable spaces
本文研究 Scarborough-Stone 问题的一个变体，探讨可分且序列紧的空间 X 的幂 X^κ 是否总是可数紧。
The Banach-Mazur problem, which asks if every infinite-dimensional Banach space has an infinite-dimensional separable quotient space, has remained unsolved for 85 years, but has been answered in the affirmative for special cases such as reflexive Banach spaces. It is also known that every infinite-dimensional non-normable Fréchet space has an infinite-dimensional separable quotient space, namely $\mathbb{R}^ω$. It is proved in this paper that every infinite-dimensional Fréchet space (including every infinite-dimensional Banach space), indeed every locally convex space which has a subspace whic…
The main purpose of this paper is to study \emph{$e$-separable spaces}, originally introduced by Kurepa as $K_0'$ spaces; we call a space $X$ $e$-separable iff $X$ has a dense set which is the union of countably many closed discrete sets. We primarily focus on the behaviour of $e$-separable spaces under products and the cardinal invariants that are naturally related to $e$-separable spaces. Our main results show that the statement "there is a product of at most $\mathfrak c$ many $e$-separable spaces that fails to be $e$-separable'" is equiconsistent with the existence of a weakly compact card…
Let X be a separable metric space and let βbe the strict topology on the space of bounded continuous functions on X, which has the space of τ-additive Borel measures as a continuous dual space. We prove a Banach-Dieudonneé type result for the space of bounded continuous functions equipped with β. As a consequence, this space is hypercomplete and a Pták space. Additionally, the closed graph, inverse mapping and open mapping theorems holds for linear maps between space of this type.
We study the class of compact spaces that appear as structure spaces of separable Banach lattices. In other words, we analyze what $C(K)$ spaces appear as principal ideals of separable Banach lattices. Among other things, it is shown that every such compactum $K$ admits a strictly positive regular Borel measure of countable type that is analytic, and in the nonmetrizable case these compacta are saturated with copies of $β\mathbb N$. Some natural questions about this class are left open.
We compare three notions of genericity of separable metric structures. Our analysis provides a general model theoretic technique of showing that structures are generic in descriptive set theoretic (topological) sense and in measure theoretic sense. In particular, it gives a new perspective on Vershik's theorems on genericity and randomness of Urysohn's space among separable metric spaces.
In this paper we study the behaviour of selective separability properties in the class of Frechét-Urysohn spaces. We present two examples, the first one given in ZFC proves the existence of a countable Frechét-Urysohn (hence $R$-separable and selectively separable) space which is not $H$-separable; assuming $\mathfrak{p}=\mathfrak{c}$, we construct such an example which is also zero-dimensional and $α_{4}$. Also, motivated by a result of Barman and Dow stating that the product of two countable Frechét-Urysohn spaces is $M$-separable under PFA, we show that the MA is not sufficient here. In the…
We strengthen the property $Δ$ of a function $f:[ω_2]^2\rightarrow [ω_2]^{\leq ω}$ considered by Baumgartner and Shelah. This allows us to consider new types of amalgamations in the forcing used by Rabus, Juhász and Soukup to construct thin-very tall compact scattered spaces. We consistently obtain spaces $K$ as above where $K^n$ is hereditarily separable for each $n\in\N$. This serves as a counterexample concerning cardinal functions on compact spaces as well as having some applications in Banach spaces: the Banach space $C(K)$ is an Asplund space of density $\aleph_2$ which has no Fréchet sm…
Function spaces on separable compact lines
在鲍姆加特纳公理BA假设下，对权重为ω₁的可分紧致线K赋予点态收敛拓扑的实值连续函数空间C_p(K)进行同构分类，表明此类K恰有两个线性同胚的函数空间C_p(K)，还构造了权重为2^ω的可分紧致线K的例子。
Polish spaces of separable Banach lattices
研究可分巴拿赫格类的描述复杂性，引入两种波兰空间编码，证明相关空间是波兰子空间，还证明理想编码上的弗雷姆林射影张量积运算的可测性及图的性质。
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