$δ$的单调版本-正态性

Monotone Normality and Nabla-Products

Roitman's combinatorial principle $Δ$ is equivalent to monotone normality of the nabla product, $\nabla (ω+1)^ω$. If $\{ X_n : n\in ω\}$ is a family of metrizable spaces and $\nabla_n X_n$ is monotonically normal, then $\nabla_n X_n$ is hereditarily paracompact. Hence, if $Δ$ holds then the box product $\square (ω+1)^ω$ is paracompact. Large fragments of $Δ$ hold in $\mathsf{ZFC}$, yielding large subspaces of $\nabla (ω+1)^ω$ that are `really' monotonically normal. Countable nabla products of metrizable spaces which are respectively: arbitrary, of size $\le \mathfrak{c}$, or separable, are mon…

可分解性与单调正态性

Resolvability and monotone normality

空间$X$被称为是$κ$ -可分解的（或几乎$κ$ -可分解的），如果它包含成对不相交的$κ$稠密集（分别是在无处稠密子集的理想上几乎不相交）。我们证明了每个拥挤单调正规（简称MN）空间都是$Δ(X)$ -可分解的和几乎$μ$ -可分解的，其中$μ= \min\{2^ω, ω_2 \}$ .另一方面，如果$κ$是可测基数，则存在MN空间$X$，其中$Δ(X) = κ$使得没有$X$的子空间是$ω_1$ -可分解的。任何基数为$< \aleph_ω$的MN空间都是最大可分解的。但从超紧基数出发，我们得到了MN空间$X$与$|X| = Δ(X) = \aleph_ω$存在的一致性，使得$X$的任何子空间都不是$ω_2$ -可分解的.

可数仿紧性的单调形式

Monotone Versions of Countable Paracompactness

可数仿紧性的一个可能的自然单调版本MCP，有一些有趣的性质。我们研究了可数仿紧性的其他各种可能的单调化以及它们之间的关系。

Countable Normality

A. V. Arhangel'skiĭ introduced in 2012, when he was visiting the department of Mathematics at King Abduaziz University, new weaker versions of normality, called \it $C$-normality, \rm and \it countable normality. \rm The purpose of this paper is to investigate countable normality property. We prove that normality implies countable normality but the converse is not true in general. We present some examples to show relationships between countable normality and other weaker versions of normality such as $C$-normality, $L$-noramlity, and mild normality. We answer the following open problem of Arha…

Monotone Covering Properties defined by Closure-Preserving Operators

We continue Gartside, Moody, and Stares' study of versions of monotone paracompactness. We show that the class of spaces with a monotone closure-preserving open operator is strictly larger than those with a monotone open locally-finite operator. We prove that monotonically metacompact GO-spaces have a monotone open locally-finite operator, and so do GO-spaces with a monotone (open or not) closure-preserving operator, whose underlying LOTS has a $σ$-closed-discrete dense subset. A GO-space with a $σ$-closed-discrete dense subset and a monotone closure-preserving operator is metrizable. A compac…

关于相对双拓扑和拓扑性质的理论

On the Theory of Relative Bitopological and Topological Properties

在工作的第一部分（第2-6节）中，特别关注相对分离公理和相对连通性，特别是讨论了p-T_0、p-T_1、p-T_2、（i，j）-和p-正则性、（i，j）-和p-完全正则性、p-实正规性和p-正规性的许多相对版本。此外，还引入和研究了（i，j）-紧性和p-紧性的相关性质，包括（i，j）-仿紧性和p-仿紧性，（i，j）-Lindeofness和p-Lindeofness，（i，j）-伪紧性和p-伪紧性的相关形式.第二部分（第7-12节）致力于，一方面，相对双拓扑归纳和覆盖维数功能，另一方面，相对版本的贝尔空间的拓扑和双拓扑的情况。最后，请注意，相对（双）拓扑性质不仅在各自理论的发展中起着特殊的作用，而且在加强以前已知的结果。