On homogeneity of Cantor cubes

可数稠密齐次与康托尔集

Countable dense homogeneity and the Cantor set

证明了CH蕴含着存在一个紧的T2空间，该空间是可数的、稠密的、齐次的、拥挤的且不包含康托尔集的拓扑副本.这与以前的结果，作者说，任何拥挤的T2空间$X$可数$π$ -重量，如果${}^ω{X}$是可数密集齐次，然后$X$必须包含一个拓扑副本的康托尔集。

Extending homeomorphisms on Cantor cubes

We discuss the question of extending homeomorphism between closed subsets of the Cantor discontinuum $D^τ$. It is established that any homeomorphism $f$ between two closed subsets of $D^τ$ can be extended to an autohomeomorphism of $D^τ$ provided $f$ preserves the $λ$-interiors of the sets for every cardinal $λ$.

广义Cantor流形与齐性

Generalized Cantor manifolds and homogeneity

Alexandroff的一个经典定理指出，每个$n$维紧集$X$包含一个$n$维Cantor流形.这个定理有许多由不同的作者得到的推广。我们考虑强Cantor流形、Mazurkiewicz流形和$V^n$ -连续体的扩张维和无限维类似物，并证明了上述定理的相应版本。我们应用我们的结果表明，每一个齐次度量化连续，这是不是在一个给定的类$\mathcal C$是一个强康托流形（或至少是康托流形）关于$\mathcal C$。这里，类$\mathcal C$是根据维度类不变量定义的四个类之一。还考虑了一类具有满足某些特殊条件的邻域基的空间。

Null sets and combinatorial covering properties

A subset of the Cantor cube is null-additive if its algebraic sum with any null set is null. We construct a set of cardinality continuum such that: all continuous images of the set into the Cantor cube are null-additive, it contains a homeomorphic copy of a set that is not null-additive, and it has the property $γ$, a strong combinatorial covering property. We also construct a nontrivial subset of the Cantor cube with the property $γ$ that is not null additive. Set-theoretic assumptions used in our constructions are far milder than used earlier by Galvin--Miller and Bartoszyński--Recław, to ob…

Cantor 立方体中的错误识别

Error recognition in the Cantor cube

本文基于薄集概念研究无限单消息传输协议，利用 Banach-Mazur 博弈证明检测错误协议为 Baire 空间且一般协议非 Borel 集非贫集，并证明 Cantor 立方体可分解为两个薄集，作为奇偶校验位的无限对应物。

On the Cantor and Hilbert Cube Frames and the Alexandroff-Hausdorff Theorem

The aim of this work is to give a pointfree description of the Cantor set. It can be shown that the Cantor set is homeomorphic to the $p$-adic integers $\mathbb{Z}_{p}:=\{x\in\mathbb{Q}_{p}: |x|_p\leq 1\}$ for every prime number $p$. To give a pointfree description of the Cantor set, we specify the frame of $\mathbb{Z}_{p}$ by generators and relations. We use the fact that the open balls centered at integers generate the open subsets of $\mathbb{Z}_{p}$ and thus we think of them as the basic generators; on this poset we impose some relations and then the resulting quotient is the frame of the …