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Constructions of thin-tall Boolean spaces. (English) Zbl 1087.54016

This paper is an expository paper on locally compact \(T_2\) scattered spaces (LCS-spaces). In the introduction, a brief, well written history on the subject is given. If the height of an LCS-space \(X\) is \(\alpha\) and the width of \(X\) is \(\beta\), \(X\) is said to be an \((\alpha,\beta)\)-LCS-space. The object of this paper is to give some methods of construction of \((\alpha,\beta)\)-LCS-spaces. Of course, all spaces constructed are in ZFC. In the first section, constructions of \((\omega,\omega_1)\)-LCS-space and for nonzero ordinals \(\alpha<\omega_2\), \((\omega,\alpha)\)-LCS spaces, are given. The next step is the existence of \((\omega,\omega_2)\)-LCS spaces. Its answer is the Baumgartner-Shelah result: \(\text{Con}(\text{ZFC})\to \text{Con}(\text{ZFC}+\) there is an \((\omega,\omega_2)\)-LCS space) and its proof. On the construction of height greater than \(\omega_2\), the following result is obtained: \(\text{Con}(\text{ZFC})\to \text{Con}(\text{FZC}+\) for every \(\alpha<\omega_3\) there is an \((\omega,\alpha)\)-LCS-space). On the construction of uncountable weight, a general theorem is proved, but two corollaries are mentioned in familiar terminologies, namely if \(V= L\) holds, then there is a \((\kappa,\kappa^+)\)-LCS-space for every regular cardinal \(\kappa\). This result is also implied from the hypothesis that there do not exist inaccessible cardinals in ZFC.

MSC:

54G12 Scattered spaces
54A25 Cardinality properties (cardinal functions and inequalities, discrete subsets)
54A35 Consistency and independence results in general topology
03E05 Other combinatorial set theory
06E15 Stone spaces (Boolean spaces) and related structures
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