Julian Dontchev, Maximilian Ganster
The aim of this paper is to continue the study of sg-compact spaces, a topological notion much stronger than hereditary compactness. We investigate the relations between sg-compact and $C_2$-spaces and the interrelations to hereditarily sg-closed sets.
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We construct a family of Hausdorff spaces such that every finite product of spaces in the family (possibly with repetitions) is CLP-compact, while the product of all spaces in the family is non-CLP-compact. Our example will yield a single Hausdorff space $X$ such that every finite power of $X$ is CLP-compact, while no infinite power of $X$ is CLP-compact. This answers a question of Steprāns and Šostak.
The aim of this paper is to continue the study of sg-compact spaces. The class of sg-compact spaces is a proper subclass of the class of hereditarily compact spaces. In our paper we shall consider sg-compactness in product spaces. Our main result says that if a product space is sg-compact, then either all factor spaces are finite, or exactly one factor space is infinite and sg-compact and the remaining ones are finite and locally indiscrete.
In this paper we consider the problem of characterization of topological spaces that embed into countably compact Hausdorff spaces. We study the separation axioms of subspaces of countably compact Hausdorff spaces and construct an example of a regular separable scattered topological space which cannot be embedded into an Urysohn countably compact topological space but embeds into a Hausdorff countably compact space.
Inspired by a recent work of Dias and Tall, we show that a compact indestructible space is sequentially compact. We also prove that a Lindelof Hausdorff indestructible space has the finite derived set property and a compact Hausdorff indestructible space is pseudoradial.
In \cite{Chaber}, Chaber has proved that countably compact spaces with a quasi $G_{δ}$-diagonal are compact. We prove that initially $κ$% -compact spaces with a quasi $G_{κ}$-diagonal are compact, for any infinite cardinal $κ.
On T-locally compact spaces
本文引入并初步研究 T-局部紧空间，阐明其与局部紧性相互独立，证明 Hausdorff 局部紧空间满足该性质，且其为拓扑性质但不被积拓扑保持。
A topological measure on a locally compact space is a set function on open and closed subsets which is finitely additive on the collection of open and compact sets, inner regular on open sets, and outer regular on closed sets. Almost all works devoted to topological measures, corresponding non-linear functionals, and their applications deal with compact spaces. The present paper is one in a series that investigates topological measures and corresponding non-linear functionals on locally compact spaces. Here we examine solid and semi-solid sets on a locally compact space. We then give a method …
A theorem by Norman L. Noble from 1970 asserts that every product of completely regular, locally pseudo-compact k_R-spaces is a k_R-space. As a consequence, all direct products of locally compact Hausdorff spaces are k_R-spaces. We provide a streamlined proof for this fact.
Under the continuum hypothesis, there is a compact homogeneous strong S-space.
We introduce the open degree of a compact space, and we show that for every natural number n, the separable Rosenthal compact spaces of degree n have a finite basis.
We show that a linearly ordered topological space is initially λ-compact if and only if it is λ-bounded, that is, every set of cardinality $\leq λ$ has compact closure. As a consequence, every product of initially λ-compact linearly ordered topological spaces is initially λ-compact.
A new method for constructing a compact and non SI-compact space
本文提出构造紧且非SI紧空间的新方法，引入v紧性概念并证明其相关性质，解答了Zhao、Ho及[1]提出的问题。
On $σ$-compact Hattori spaces
本文研究σ-紧Hattori空间，给出其若干等价刻画，并排除了一些不成立的候选刻画。
For a separable locally compact but not compact metrizable space $X$, let $αX = X \cup \{x_\infty\}$ be the one-point compactification with the point at infinity $x_\infty$. We denote by $EM(X)$ the space consisting of admissible metrics on $X$, which can be extended to an admissible metric on $αX$, endowed with the compact-open topology. Let $ \mathbf{c}_0 \subset (0,1)^\mathbb{N}$ be the space of sequences converging to $0$. In this paper, we shall show that if $X$ is separable, locally connected and locally compact but not compact, and there exists a sequence $\{C_i\}$ of connected sets in …
Hurewicz proved completely metrizable Menger spaces are /sigma-compact. We extend this to Cech-complete Menger spaces and consistently to projective Menger metrizable spaces. On the other hand, it is consistent that there is a co-analytic Menger space that is not /sigma-compact.
We continue to investigate applications of $k$-covers in function spaces with the compact-open topology.
We investigate certain geometric properties of the spaces of idempotent measures. In particular, we prove that the space of idempotent measures on an infinite compact metric space is homeomorphic to the Hilbert cube.
We prove that every fragmentable linearly ordered compact space is almost totally disconnected. This combined with a result of Arvanitakis yields that every linearly ordered quasi Radon-Nikodym compact space is Radon-Nikodym, providing a new partial answer to the problem of continuous images of Radon-Nikodym compacta.
Our work aims to introduce generalization of soft $ μ$-compact soft generalized topological spaces, namely; soft nearly $ μ$-compact spaces which are defined over initial universe with a fixed set of parameters. Basic properties and some significant corollaries will be introduced, Moreover, we investigated that a soft nearly $ μ$-compact space produces a parametrized family of nearly $ μ$-compact spaces. Some counter examples will be established to show that the converse may not be hold.
A topological preordered space admits a Hausdorff closed preorder compactification if and only if it is Tychonoff and the preorder is represented by the family of continuous isotone functions. We construct the largest Hausdorff closed preorder compactification for these spaces and clarify its relation with Nachbin's compactification. Under local compactness the problem of the existence and identification of the smallest Hausdorff closed preorder compactification is considered.
As it was introduced by Tkachuk and Wilson, a topological space $X$ is cellular-compact if given any cellular, i.e. disjoint, family $\mathcal U$ of non-empty open subsets of $X$ there is a compact subspace $K\subset X$ such that $K\cap U\ne \emptyset$ for each $U\in \mathcal U$. Answering several questions raised by Tkachuk and Wilson we show that (1) any first countable cellular-compact $T_2$ space is $T_3$, and so its cardinality is at most $\mathfrak{c} = 2^ω$; (2) $cov(\mathcal M)>ω_1$ implies that every first countable and separable cellular-compact $T_2$ space is compact; (3 if there is…
We give a new characterization of Valdivia compact spaces: A compact space is Valdivia if and only if it has a dense commutatively monotonically retractable subspace. This result solves Problem 5.12 from \cite{sal-rey}. Besides, we introduce the notion of full $c$-skeleton and prove that a compact space is Corson if and only if it has a full $c$-skeleton.
We give a definition of compactness in L-fuzzy topological spaces and provide a characterization of compact L-fuzzy topological spaces, where L is a complete quasi-monoidal lattice with some additional structures, and we present a version of Tychonoff's theorem within the category of L-fuzzy topological spaces.
We characterize the finite dimensional asymmetric normed spaces which are right bounded and the relation of this property with the natural compactness properties of the unit ball, as compactness and strong compactness. In contrast with some results found in the existing literature, we show that not all right bounded asymmetric norms have compact closed balls. We also prove that there are finite dimensional asymmetric normed spaces that satisfy that the closed unit ball is compact, but not strongly compact, closing in this way an open question on the topology of finite dimensional asymmetric no…
We prove some consistency results concerning the Moving Off Property for locally compact spaces and thus the question of whether their function spaces are Baire.
In this paper we examine two basic topological properties of partial metric spaces, namely compactness and completeness. Our main result claims that in these spaces compactness is equivalent to sequential compactness. We also show that Hausdorff compact partial metric spaces are metrizable. In the second part of this article we discuss the significance of bottom sets of partial metric spaces in fixed point theorems for mappings acting in these spaces.
We continue studying the properties of $γ_0$-compact, $γ^*$-regular and $γ$-normal spaces defined in [5]. We also define and discuss $γ$-locally compact spaces.
We provide examples of nonseparable compact spaces with the property that any continuous image which is homeomorphic to a finite product of spaces has a maximal prescribed number of nonseparable factors.
Two problems of SI-compact spaces
本文针对SI-紧空间的两个问题，证明有限个SI-紧空间的乘积仍为SI-紧空间，正面回应了前人的问题，并得出SI-紧空间范畴不是TOP的反射满子范畴的结论。
In the paper, we investigate (scattered) compact spaces with a $P$-base for some poset $P$. More specifically, we prove that, under the assumption $ω_1<\mathfrak{b}$, any compact space with an $ω^ω$-base is first-countable and any scattered compact space with an $ω^ω$-base is countable. These give positive solutions to Problems 8.6.9 and 8.7.7 in \cite{Banakh2019}. Using forcing, we also prove that in a model of $ω_1<\mathfrak{b}$, there is a non-first-countable compact space with a $P$-base for some poset $P$ with calibre~$ω_1$.
The aim of this paper is to prove the existence of inductive and inverse limits of direct and inverse systems in a certain category of compact metric spaces as well as of compact metric groups. Some applications are presented.
We prove that if K is a remainder of the Hilbert space (i.e., K is the complement of the Hilbert space in its metrizable compactification) then every non-one-point closed image of K either contains a compact set with no transfinite dimension or contains compact sets of arbitrarily high inductive transfinite dimension ind. We construct also for each natural n a sigma-compact metrizable n-dimensional space whose image under any non-constant closed map has dimension at least n, and analogous examples for the transfinite dimension ind.
We present a connected version of the compact $L$-space constructed by Kenneth Kunen under CH. We show that this provides a Corson compact space $K$ such that the Banach space $C(K)$ is isomorphic to no space of continuous function on a zero-dimensional compactum.
In this paper, some features of countably $α$-compact topological spaces are presented and proven. The connection between countably $α$% -compact, Tychonoff, and $α$-Hausdorff spaces is explained. The space is countably $α$-compact space iff every locally finite family of non-empty subsets of such space is finite is demonstrated. The countably $% α$-compact space with weight greater than or equal to $\aleph_0$ is the $α$-continuous image of a closed subspace of the cube $D^{\aleph_0}$ is discussed. The boundedness of $α$-continuous functions mapping $α$% -compact spaces to other spaces is clea…
A compact space is said to be weakly Radon-Nikodým if it is homeomorphic to a weak*-compact subset of the dual of a Banach space not containing an isomorphic copy of $\ell_1$. In this work we provide an example of a continuous image of a Radon-Nikodým compact space which is not weakly Radon-Nikodým. Moreover, we define a superclass of the continuous images of weakly Radon-Nikodým compact spaces and study its relation with Corson compacta and weakly Radon-Nikodým compacta.
We present a construction of a compact connected space which supports a normal probability measure.
We use lexicographic products to give examples of compact spaces of first Baire class functions on a compact metric space that cannot be represented as spaces of functions with countably many discontinuities.
An $ω_1$-compact space is a space in which every closed discrete subspace is countable. We give various general conditions under which a locally compact, $ω_1$-compact space is $σ$-countably compact, i.e., the union of countably many countably compact spaces. These conditions involve very elementary properties. Many results shown here are independent of the usual (ZFC) axioms of set theory, and the consistency of some may involve large cardinals. For example, it is independent of the ZFC axioms whether every locally compact, $ω_1$-compact space of cardinality $\aleph_1$ is $σ$-countably compac…
We show that for every Dugundji compact $K$ of weight aleph one the Banach space $C(K)$ is 1-Plichko and the space $P(K)$ of probability measures on $K$ is Valdivia compact. Combining this result with the existence of a non-Valdivia compact group, we answer a question of Kalenda.
The notions of compactness and Hausdorff separation for generalized enriched categories allow us, as classically done for the category $\mathsf{Top}$ of topological spaces and continuous functions, to study $\textit{compactly generated spaces}$ and $\textit{quasi-spaces}$ in this setting. Moreover, for a class $\mathcal{C}$ of objects we generalize the notion of $\textit{$\mathcal{C}$-generated spaces}$, from which we derive, for instance, a general concept of $\textit{Alexandroff spaces}$. Furthermore, as done for $\mathsf{Top}$, we also study, in our level of generality, the relationship bet…
A study on delta compact spaces
本文引入δ紧空间概念并研究其基本性质及在连续映射下的表现，同时提出局部δ紧性概念。
There is a locally compact Hausdorff space of weight aleph_omega which is linearly Lindelof and not Lindelof. This improves an earlier result, which produced such a space of weight beth_omega.
A convex subset X of a linear topological space is called compactly convex if there is a continuous compact-valued map $Φ:X\to exp(X)$ such that $[x,y]\subsetΦ(x)\cup Φ(y)$ for all $x,y\in X$. We prove that each convex subset of the plane is compactly convex. On the other hand, the space $R^3$ contains a convex set that is not compactly convex. Each compactly convex subset $X$ of a linear topological space $L$ has locally compact closure $\bar X$ which is metrizable if and only if each compact subset of $X$ is metrizable.
For metrizable spaces we replace the notion of almost periodic homeomorphism with a similar notion and verify that the usual characterizations of almost periodic homeomorphisms of compact metric spaces are valid for all metrizable spaces.
We prove that compact Hausdorff spaces with a $\mathbb{P}$-diagonal are metrizable.
This is a detailed introductory survey of the cohomological dimension theory of compact metric spaces.
We prove that a continuous image of a Radon-Nikodým compact space of weight less than b is Radon-Nikodým compact. As a Banach space counterpart, subspaces of Asplund generated Banach spaces of density character less than b are Asplund generated. In this case, in addition, there exists a subspace of an Asplund generated space which is not Asplund generated which has density character exactly b.
We discuss which groups can be realized as the fundamental groups of compact Hausdorff spaces. In particular, we prove that the claim ``every group can be realized as the fundamental group of a compact Hausdorff space'' is consistent with the Zermelo - Fraenkel - Choice set theory.
A duality theorem for the category of locally compact Hausdorff spaces and continuous maps which generalizes the well-known Duality Theorem of de Vries is proved.
We consider the cardinal sequences of compact scattered spaces in models where CH is false. We describe a number of models where the continuum is aleph_2 in which no such space can have aleph_2 countable levels.
Homogeneous locally compact spaces
本文综述局部紧齐性度量空间的最新结果与未解决问题，重点讨论齐性有限维ANR-空间。
Let $X$ be a zero-dimensional metric space and $X'$ its derived set. We prove the following assertions: (1) the space $C_k(X,2)$ is an Ascoli space iff $C_k(X,2)$ is $k_\mathbb{R}$-space iff either $X$ is locally compact or $X$ is not locally compact but $X'$ is compact, (2) $C_k(X,2)$ is a $k$-space iff either $X$ is a topological sum of a Polish locally compact space and a discrete space or $X$ is not locally compact but $X'$ is compact, (3) $C_k(X,2)$ is a sequential space iff $X$ is a Polish space and either $X$ is locally compact or $X$ is not locally compact but $X'$ is compact, (4) $C_k…
We begin the study of the consequences of the existence of certain infinite matrices. Our present application is to compactness of products of topological spaces.
It is a well known open problem if, in ZFC, each compact space with a small diagonal is metrizable. We explore properties of compact spaces with a small diagonal using elementary chains of submodels. We prove that ccc subspaces of such spaces have countable π-weight. We generalize a result of Gruenhage about spaces which are metrizably fibered. Finally we discover that if there is a Luzin set of reals, then every compact space with a small diagonal will have many points of countable character.
In the context of classical associations between classes of Banach spaces and classes of compact Hausdorff spaces we survey known results and open questions concerning the existence and nonexistence of universal Banach spaces and of universal compact spaces in various classes. This gives quite a complex network of interrelations which quite often depend on additional set-theoretic assumptions.
We show that the direct sum of uncountably many non-Abelian groups does not embed into the group of homeomorphisms of a compact metric space.
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