权重、净重和基本子模型

IF 0.5 4区数学 Q3 MATHEMATICS

Topology and its Applications Pub Date : 2025-06-02 DOI:10.1016/j.topol.2025.109469

Alan Dow , István Juhász

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引用次数: 0

摘要

在这篇文章中，我们证明了与[6]的一些结果和问题有关的几个定理。我们回答b[6]中提出的两个主要问题。首先，我们给出一个在C（ω1）中具有不可数净重的Hausdorff空间的ZFC例子。然后我们证明在向CH的模型中加入任意数量的Cohen实数后，在扩展中C（ω1）中的每一个正则空间都有可数的净权。在最后一节中，我们在ZFC中证明了以下两个命题：(i)如果S∧ω1是平稳的，那么对于S上权值不可数的任意正则拓扑 S也有一个权值不可数的非平稳子集。（ii）对于ω1上的任意拓扑，如果ω1的所有末段都有不可数的权值，则ω1有一个权值不可数的非平稳子集。与此相反，[6]中表明，净重的类似表述在ZFC中是不可证明的。值得注意的是，我们对上述结果的所有证明都必不可少地使用了基本子模型。

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Weight, net weight, and elementary submodels

In this note we prove several theorems that are related to some results and problems from [6].

We answer two of the main questions that were raised in [6]. First we give a ZFC example of a Hausdorff space in

C (ω_{1})

that has uncountable net weight. Then we prove that after adding any number of Cohen reals to a model of CH, in the extension every regular space in

C (ω_{1})

has countable net weight.

In the last section we prove in ZFC the following two statements:

(i) If

S \subset ω_{1}

is stationary then for any regular topology on S of uncountable weight  S has a non-stationary subset that has uncountable weight as well.

(ii) For any topology on

ω_{1}

, if all final segments of

ω_{1}

have uncountable weight then

ω_{1}

has a non-stationary subset of uncountable weight.

In contrast to this, it was shown in [6] that the analogous statements for net weight are not provable in ZFC.

It is remarkable that all our proofs of the above results make essential use of elementary submodels.

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来源期刊

Topology and its Applications 数学-数学

CiteScore

1.20

自引率

33.30%

发文量

251

审稿时长

6 months

期刊介绍： Topology and its Applications is primarily concerned with publishing original research papers of moderate length. However, a limited number of carefully selected survey or expository papers are also included. The mathematical focus of the journal is that suggested by the title: Research in Topology. It is felt that it is inadvisable to attempt a definitive description of topology as understood for this journal. Certainly the subject includes the algebraic, general, geometric, and set-theoretic facets of topology as well as areas of interactions between topology and other mathematical disciplines, e.g. topological algebra, topological dynamics, functional analysis, category theory. Since the roles of various aspects of topology continue to change, the non-specific delineation of topics serves to reflect the current state of research in topology. At regular intervals, the journal publishes a section entitled Open Problems in Topology, edited by J. van Mill and G.M. Reed. This is a status report on the 1100 problems listed in the book of the same name published by North-Holland in 1990, edited by van Mill and Reed.