Kurepa trees and topological non-reflection

Abstract

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 κ, we say that P does not reflect at κ. We undertake a fine analysis of Kurepa trees which results in defining canonical topological and combinatorial structures associated with the tree which possess a remarkably wide range of nonreflecting properties providing new constructions and solutions of open problems in topology. The most interesting results show that many known properties may not reflect at any fixed singular cardinal of uncountable cofinality. The topological properties we consider vary from normality, collectionwise Hausdorff property to metrizablity and many others. The combinatorial properties are related to stationary reflection.

On constructions with 2-cardinals

“…So, often the stationary nonreflection is the underlying one, hence in this context it is natural to consider stationary stepping up tools. (see [51], [24], [30] or in the κ context e.g. [14]).…”

On properties of compacta that do not reflect in small continuous images

“…The corollary above gives partial negative answers to problems posed by Jardón and Tkachuk ([10], Questions 4. [13][14][15] if we assume the continuum hypothesis together with 3.1, so for instance if we are in the constructible universe.…”

“…There is a significant amount of research related to properties of structures that reflect in substructures of smaller cardinality, see e.g. Bagaria, Magidor, Sakai [3], Koszmider [13,14], Fuchino and Rinot [9], Tall [23]. Reflection phenomena in topology are usually studied according to the following pattern: Problem 1.1.…”

“…In addition, our space K gives partial negative answers to problems posed by Jardón and Tkachuk ([10], Questions 4. [13][14][15] on the reflection of type 1.1 for Corson compacta and related classes. The construction of the space is given in section 3 and uses the familiar idea of a ladder system associated to a given set S ⊆ ω 2 ; see, for instance, Pol [22] and Ciesielski and Pol [6] where a construction of this type was used to solve some problems on the structure of C(K) spaces.…”