W. Kubiś, A. Leiderman
We investigate the class of compact spaces which are embeddable into a power of the real line $R^κ$ in such a way that c_0(κ) is dense in the image. We show that this is a proper subclass of the class of Valdivia, even when restricted to Corson compacta. We prove a preservation result concerning inverse sequences with semi-open retractions. As a corollary we obtain that retracts of Cantor or Tikhonov cubes belong to the above class.
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We study properties of semi-Eberlein compacta related to inverse limits. We concentrate our investigation on an interesting subclass of small semi-Eberlein compacta whose elements are obtained as inverse limits whose bonding maps are semi-open retractions.
We discuss two problems concerning the class Eberlein compacta, i.e., weakly compact subspaces of Banach spaces. The first one deals with preservation of some classes of scattered Eberlein compacta under continuous images. The second one concerns the known problem of the existence of nonmetrizable compact spaces without nonmetrizable zero-dimensional closed subspaces. We show that the existence of such Eberlein compacta is consistent with ZFC. We also show that it is consistent with ZFC that each Eberlein compact space of weight $> ω_1$ contains a nonmetrizable closed zero-dimensional subspace…
We deeply study retractions associated to suitable models in compact spaces admitting a retractional skeleton and find several interesting consequences. Most importantly, we provide a new characterization of Valdivia compacta using the notion of retractional skeletons, which seems to be helpful when characterizing its subclasses. Further, we characterize Eberlein and semi-Eberlein compacta in terms of retractional skeletons and show that our new characterizations give an alternative proof of the fact that continuous image of an Eberlein compact is Eberlein as well as new stability results for …
A space $X$ is Eberlein-Grothendieck if $X\subset C_p(K)$ for some compact space $K.$ In this paper we address the problem of whether such a space $X$ is $σ$-discrete whenever it is scattered. We show that if $w(K)\leqω_1$ then $X$ is $σ$-dicrete whenever $X$ has height $ω_1$ and it is locally compact or locally countable. It is also proved that every Lindelöf Čech-complete scattered space is $σ$-compact.
We present a characterization of spaces of strictly decreasing functions on trees in terms of bisequentiality. This characterization answers Questions 6.1 and 6.2 of "A filter on a collection of finite sets and Eberlein compacta" by T. Cieśla. Moreover we study the relation between these spaces and the classes of Corson, Eberlein and uniform Eberlein compacta.
We prove that each metrizable space (of cardinality less or equal to continuum) has a (first countable) uniform Eberlein compactification and each scattered metrizable space has a scattered hereditarily paracompact compactification. Each compact scattered hereditarily paracompact space is uniform Eberlein and belongs to the smallest class of compact spaces, that contain the empty set, the singleton, and is closed under producing the Aleksandrov compactification of the topological sum of a family of compacta from that class.
For k being the first uncountable cardinal w_1 or k being the cardinality of the continuum c, we prove that it is consistent that there is no Banach space of density k in which it is possible to isomorphically embed every Banach space of the same density which has a uniformly Gâteaux differentiable renorming or, equivalently, whose dual unit ball with the weak* topology is a subspace of a Hilbert space (a uniform Eberlein compact space). This complements a consequence of results of M. Bell and of M. Fabian, G. Godefroy, V. Zizler that assuming the continuum hypothesis, there is a universal spa…
We work in set-theory without choice $\ZF$. Given a closed subset $F$ of $[0,1]^I$ which is a bounded subset of $\ell^1(I)$ ({\em resp.} such that $F \subseteq \ell^0(I)$), we show that the countable axiom of choice for finite subsets of $I$, ({\em resp.} the countable axiom of choice $\ACD$) implies that $F$ is compact. This enhances previous results where $\ACD$ ({\em resp.} the axiom of Dependent Choices $\DC$) was required. Moreover, if $I$ is linearly orderable (for example $I=\IR$), the closed unit ball of $\ell^2(I)$ is weakly compact (in $\ZF$).
It follows from a theorem of Rosenthal that a compact space is $ccc$ if and only if every Eberlein continuous image is metrizable. Motivated by this result, for a class of compact spaces $\mathcal{C}$ we define its orthogonal $\mathcal{C}^\perp$ as the class of all compact spaces for which every continuous image in $\mathcal{C}$ is metrizable. We study how this operation relates classes where centeredness is scarce with classes where it is abundant (like Eberlein and $ccc$ compacta), and also classes where independence is scarce (most notably weakly Radon-Nikodým compacta) with classes where i…
Assuming that there is a stationary set in $ω_{2}$ of ordinals of countable cofinality that does not reflect, we prove that there exists a compact space which is not Corson compact and whose all continuous images of weight at most $ω_1$ are Eberlein compacta. This yields an example of a Banach space of density $ω_{2}$ which is not weakly compactly generated but all its subspaces of density $ω_{1}$ are weakly compactly generated. We also prove that under Martin's axiom countable functional tightness does not reflect in small continuous images of compacta.
We show that there are no automorphic Banach spaces of the form C(K) with K continuous image of Valdivia compact except the spaces c0(I). Nevertheless, when K is an Eberlein compact of finite height such that C(K) is not isomorphic to c0(I), all isomorphism between subspaces of C(K) of size less than aleph_omega extend to automorphisms of C(K).
In this paper some applications of the methods and results of its first part and of the results of M. Stone, H. de Vries, P. Roeper are given. In particular: some generalizations of the Stone Duality Theorem are obtained; a completion theorem for local contact Boolean algebras is proved; a direct proof of the Ponomarev's solution of Birkhoff's Problem 72 is found, and the spaces which are co-absolute with the (zero-dimensional) Eberlein compacts are described.
Given a compact space in a fixed universe of set theory, one can naturally define its interpretation in any ZFC extension of the universe. We investigate the stability of some classes of compact spaces with respect to extensions of this sort. We show that the class of Eberlein/Gul'ko compacta is stable (= absolute). On the other hand, there are examples of Corson compacta which are no longer Corson in some forcing extensions. All the material comes from the author's notes written between 2004 -- 2006.
A topological space $X$ is called hereditarily supercompact if each closed subspace of X is supercompact. By a combined result of Bula, Nikiel, Tuncali, Tymchatyn, and Rudin, each monotonically normal compact Hausdorff space is hereditarily supercompact. A dyadic compact space is hereditarily supercompact if and only if it is metrizable. Under (MA + not CH) each separable hereditarily supercompact space is hereditarily separable and hereditarily Lindelöf. This implies that under (MA + not CH) a scattered compact space is metrizable if and only if it is separable and hereditarily supercompact. …
On the class of NY compact spaces of finitely supported elements and related classes
本文通过σ-积嵌入条件给出ω-Corson与NY紧空间的新刻画，构造反例回答Kubiś和Leiderman的问题，并证明NY紧性由函数空间C_p(K)的拓扑结构决定。
Countability Properties of Weakly Compact Sets in Asymmetric Locally Convex Spaces
本文研究了不对称局部凸空间中弱紧集的可数性性质，引入了 $C_a(K)$ 空间，并探讨了不同紧性条件的等价性及 Fréchet-Urysohn 性质。
In our paper [18] we showed that a Tychonoff space $X$ is a $Δ$-space (in the sense of [20], [30]) if and only if the locally convex space $C_{p}(X)$ is distinguished. Continuing this research, we investigate whether the class $Δ$ of $Δ$-spaces is invariant under the basic topological operations. We prove that if $X \in Δ$ and $φ:X \to Y$ is a continuous surjection such that $φ(F)$ is an $F_σ$-set in $Y$ for every closed set $F \subset X$, then also $Y\in Δ$. As a consequence, if $X$ is a countable union of closed subspaces $X_i$ such that each $X_i\in Δ$, then also $X\in Δ$. In particular, $σ…
Grothendieck's theorem on the precompactness of subsets functional spaces over pseudocompact spaces
本文推广Eberlein和Grothendieck定理，证明在伪紧空间含稠密Lindelöf Σ-空间时，函数空间C_p(X)的伪紧子空间预紧，并给出Cech完备乘积情形下有界子集预紧及分离连续函数连续性结果。
Denote by $[0,ω_1)$ the locally compact Hausdorff space consisting of all countable ordinals, equipped with the order topology, and let $C_0[0,ω_1)$ be the Banach space of scalar-valued, continuous functions which are defined on $[0,ω_1)$ and vanish eventually. We show that a weakly$^*$ compact subset of the dual space of $C_0[0,ω_1)$ is either uniformly Eberlein compact, or it contains a homeomorphic copy of the ordinal interval $[0,ω_1]$. Using this result, we deduce that a Banach space which is a quotient of $C_0[0,ω_1)$ can either be embedded in a Hilbert-generated Banach space, or it is i…
The aim of this paper is to clarify the properties of semi-barrelled spaces (also called countably quasi-barrelled spaces in the literature). These spaces were studied by several authors, in particular in the classical book of N. Bourbaki "Espaces vectoriels topologiques". However, six incorrect statements can be found in this reference. In particular: a Hausdorff and quasi-complete semi-barrelled space is complete, a semi-barrelled, semi-reflexive space is complete, a locally convex hull of semi-barrelled semi-reflexive spaces is semi-reflexive, a locally convex hull of semi-barrelled reflexi…
A new class of functions called L-fuzzy weakly Semi-Preopen (Semi-Preclosed) functions in L-fuzzy topological spaces are introduced in this paper. Some characterizations of this class and its properties and the relationship with other classes of functions between L-fuzzy topological spaces are also obtained.
Here we have studied the idea of semi Lamda*-closed sets and investigate some of their properties in spaces considered by A. D. Alexandroff [1]. We have introduced some new separation axioms namely semi-Tw/4, , semi-T3w/8, semi-T5w/8, and their inter-relation with semi-T0 and semi-T1. We have shown that under certain conditions these axioms are equivalent.
In this paper, we continue studying the properties of $γ^{*}$-semi-open sets in topological spaces introduced by S. Hussain, B. Ahmad and T. Noiri[8]. We also introduce and discuss the $γ^{*}$-semi-continuous functions which generalize semi-continuous functions defined by N. Levine [10].
It is proved that there is no structure of left (right) cancelative semigroup on $[L]$-dimensional universal space for the class of separable compact spaces of extensional dimension $\le [L]$. Besides, we note that the homeomorphism group of $[L]$-dimensional space whose nonempty open sets are universal for the class of separable compact spaces of extensional dimension $\le [L]$ is totally disconnected.
$Ψ$-Spaces and Semi-Proximality
本文研究了$Ψ$-空间中的半近邻性，证明了几乎不相交族和梯系统下半近邻性的特征，并得出非半近邻$Ψ$-空间的存在性独立于ZFC的结论。
Semi-coarse Spaces: Fundamental Groupoid and the van Kampen Theorem
本文构造了半粗空间上的基本广群不变量，证明了其对一般半粗空间良定义且对粗空间非平凡，并证明了其满足范坎彭定理。
We prove that the semigroup operation of a topological semigroup $S$ extends to a continuous semigroup operation on its the Stone-Čech compactification $βS$ provided $S$ is a pseudocompact openly factorizable space, which means that each map $f:S\to Y$ to a second countable space $Y$ can be written as the composition $f=g\circ p$ of an open map $p:X\to Z$ onto a second countable space $Z$ and a map $g:Z\to Y$. We present a spectral characterization of openly factorizable spaces and establish some properties of such spaces.
We study computable topological spaces and semicomputable and computable sets in these spaces. In particular, we investigate conditions under which semicomputable sets are computable. We prove that a semicomputable compact manifold $M$ is computable if its boundary $\partial M$ is computable. We also show how this result combined with certain construction which compactifies a semicomputable set leads to the conclusion that some noncompact semicomputable manifolds in computable metric spaces are computable.
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