Kurepa trees and topological non-reflection
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 \(\kappa\).
scientific article; zbMATH DE number 2174906
scientific article