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Kurepa trees and topological non-reflection. (English) Zbl 1076.54003

The author is mainly interested in a Kurepa tree. A tree is called a Kurepa tree if it has countable levels and more than \(\omega_1\) uncountable branches. The existence of a Kurepa tree is called Kurepa’s hypothesis (KH).
All trees are assumed to be of height \(\omega_1\). \(T\) is called a tree with many branches if for every \(t\) in \(T\), there is an uncountable branch of \(T\) whose range contains \(t\). \(\text{Br}(T)\) denotes the collection of all uncountable branches of \(T\).
KH\(^\lambda(T)\) denotes the following statement: There is an \(\omega_1\)-tree \(T\) with exactly \(\lambda\)-many branches and \(\lambda^\omega= \lambda\).
The main object investigated in this paper is \({\mathcal F}(T)\) – some well-founded and cofinal subfamily of \([\text{Br}(T)]^\omega\), the family of all countable collections of branches of \(T\), for any normal tree \(T\) with many branches. The topology on \({\mathcal F}(T)\) is defined by \[ {\mathcal B}(T)= \{U(\theta, F): F\in [\text{Br}(T)]^\omega, \theta< \omega_1\}\cup\{\phi\} \] as its basis, where for \(F\in[\text{Br}(T)]^{<\omega}\) and \(\theta<\omega_1\), \[ {\mathcal F}_\theta(T)= \{X\in [\text{Br}(T)]^\omega:\forall t\in \text{Lev}_\theta T\exists ! b\in X,\;t\in\text{ran}(b)\} \] and \[ U(\theta, F)= \Biggl\{X\in\bigcup_{0\leq\theta_1\leq\theta} {\mathcal F}_{\theta_1}(T): F\subset X\Biggr\}. \] The author proves the following theorems as main results:
Theorem 35. Suppose KH\(^\lambda(T)\) holds for \(\lambda>\omega_1\). Then there is a zero-dimensional, regular (weakly collectionwise Hausdorff, if \(\text{cf}(\lambda)>\omega_1\)) locally metrizable, non-\(\sigma\)-para-Lindelöf, hence, nonparacompact, nonmetrizable space \({\mathcal F}(T)\) of size \(\lambda\) such that every one of its subspaces of size less than \(\lambda\) is metrizable and included in a clopen metrizable subspace. Moreover, \({\mathcal F}(T)\) is not a generalized ordered space nor a monotonically normal space.
\(S(T)\) denotes the statement that \({\mathcal F}(T)\) is stationary in \([\text{Br}(T)]^\omega\). Theorem 42. Let \(\lambda> \omega_1\) be a cardinal of uncountable cofinality. Suppose that KH\(^\lambda(T)\) and \(S(T)\) hold, then there is a zero-dimensional, collectionwise normal, locally metrizable, countably paracompact, non-meta-Lindelöf, nonperfectly normal, nonmetrizable space of size \(\lambda\), such that every one of its subspaces of size less than \(\lambda\) is metrizable. Moreover, the space is not a monotonically normal space hence it is not a generalized ordered space.
Theorem 49. Suppose KH\(^\lambda\) holds, then there is a zerodimensional, regular (weakly collectionwise Hausdorff if \(\text{cf}(\lambda)> \omega_1\)), nonnormal, noncollectionwise Hausdorff, locally metrizable, meta-Lindelöf but non-\(\sigma\)-para-Lindelöf space of size \(\lambda\) such that any of its subspaces of size less than \(\lambda\) is metrizable and included in a clopen metrizable space (hence the space is \((<\lambda)\)-collectionwise Hausdorff). Moreover, the space is not monotonically normal and hence it is not a generalized ordered space.
The author explains the reason to use the term ‘topological nonreflection’, as below:
The author would like to present some ‘incompactness of certain natural topological properties’ that follows from KH. Since in the topological context, the notion of compactness has a different meaning he opted for the word reflection or nonreflection. A property \(P\) of a structure \(S\) does not reflect if no substructure of \(S\) of smaller cardinality than \(S\) has the property. If for a given property \(P\) there is such an \(S\) of cardinality \(\kappa\), it is said that \(P\) does not reflect at \(\kappa\).

MSC:

54A25 Cardinality properties (cardinal functions and inequalities, discrete subsets)
03E04 Ordered sets and their cofinalities; pcf theory
54D15 Higher separation axioms (completely regular, normal, perfectly or collectionwise normal, etc.)
54D20 Noncompact covering properties (paracompact, Lindelöf, etc.)
54E40 Special maps on metric spaces
54E35 Metric spaces, metrizability
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