Remarks on the space ℵ1 in ZF

AbstractWe show:(1)ℵ1 with the order topology is effectively normal, i.e., there is a function associating to every pair (A,B) of disjoint closed subsets of ℵ1 a pair (U,V) of disjoint open sets with A⊆U and B⊆V.(2)For every countable ordinal α the ordered space α is metrizable. Hence, every closed subset of α is a zero set and consequently the Čech–Stone extension of α coincides with its Wallman extension.(3)In the Feferman–Levy model where ℵ1 is singular, the ordinal space ℵ1 is base-Lindelöf but not Lindelöf.(4)The Čech–Stone extension βℵ1 of ℵ1 is compact iff its Wallman extension W(ℵ1) is compact.(5)The set L of all limit ordinals of ℵ1 is not a zero set

Chain-condition methods in topology

AbstractThe special role of countability in topology has been recognized and commented upon very early in the development of the subject. For example, especially striking and insightful comments in this regard can be found already in some works of Weil and Tukey from the 1930s (see, e.g., Weil (1938) and Tukey (1940, p. 83)). In this paper we try to expose the chain condition method as a powerful tool in studying this role of countability in topology. We survey basic countability requirements starting from the weakest one which originated with the famous problem of Souslin (1920) and going towards the strongest ones, the separability and metrizability conditions. We have tried to expose the rather wide range of places where the method is relevant as well as some unifying features of the method

ON SOME OPEN PROBLEMS IN BANACH SPACE THEORY

The main line of investigation of the present work is the study of some aspects in the analysis of the structure of the unit ball of (infinite-dimensional) Banach spaces. In particular, we analyse some questions concerning the existence of suitable renormings that allow the new unit ball to possess a specific geometric property. The main part of the thesis is, however, dedicated to results of isometric nature, in which the original norm is the one under consideration. One of the main sources for the selection of the topics of investigation has been the recent monograph [GMZ16], entirely dedicated to collecting several open problems in Banach space theory and formulating new lines of investigation. We take this opportunity to acknowledge the authors for their effort, that offered such useful text to the mathematical community. The results to be discussed in our work actually succeed in solving a few of the problems presented in the monograph and are based on the papers [H\ue1Ru17, HKR18, H\ue1Ru19, HKR\u2022\u2022]. Let us say now a few words on how the material is organised. The thesis is divided in four chapters (some whose contents are outlined below) which are essentially independent and can be read in whatsoever order. The unique chapter which is not completely independent from the others is Chapter 4, where we use some results from Chapter 2 and which is, in a sense, the non-separable prosecution of Chapter 3. However, cross-references are few (never implicit) and usually restricted to quoting some result; it should therefore be no problem to start reading from Chapter 4. The single chapters all share the same arrangement. A first section is dedicated to an introduction to the subject of the chapter; occasionally, we also present the proof of known results, in most cases as an illustration of an important technique in the area. In these introductions we strove to be as self-contained as possible in order to help the novel reader to enter the field; consequently, experts in the area may find them somewhat redundant and prefer to skip most parts of them. The first section of each chapter concludes with the statement of our most significant results and a comparison with the literature. The proofs of these results, together with additional results or generalisations, are presented in the remaining sections of the chapter. These sections usually follow closely the corresponding articles (carefully referenced) where the results were presented

Notes in Pure Mathematics & Mathematical Structures in Physics

These Notes deal with various areas of mathematics, and seek reciprocal combinations, explore mutual relations, ranging from abstract objects to problems in physics.Comment: Small improvements and addition

Analysis in univalent type theory

Some constructive real analysis is developed in univalent type theory (UTT). We develop various types of real numbers, and prove several equivalences between those types. We then study computation with real numbers. It is well known how to compute with real numbers in intensional formalizations of mathematics, where equality of real numbers is specified by an imposed equivalence relation on representations such as Cauchy sequences. However, because in UTT equality of real numbers is captured directly by identity types, we are prevented from making any nontrivial discrete observations of arbitrary real numbers. For instance, there is no function which converts real numbers to decimal expansions, as this would not be continuous. To avoid breaking extensionality, we thus restrict our attention to real numbers that have been equipped with a simple structure called a \emph{locator}. In order to compute, we modify existing constructions in analysis to work with locators, including Riemann integrals, intermediate value theorems and differential equations. Hence many of the proofs involving locators look familiar, showing that the use of locators is not a conceptual burden. We discuss the possibility of implementing the work in proof assistants and present a Haskell prototype

A nonlinear theory of generalized tensor fields on Riemannian manifolds

Diese Dissertation behandelt drei verwandte Themenbereiche im Gebiet der vollen diffeomorphismeninvarianten Colombeau'schen Algebren. Teil I umfasst eine Erweiterung der Theorie der vollen diffeomorphismeninvarianten Colombeau'schen Algebren auf den Fall von Tensorfeldern auf Riemannschen Mannigfaltigkeiten. Eine wesentliche Rolle spielt dabei der Levi-Civita-Zusammenhang mittels welchem distributionelle Tensorfelder regularisiert und somit auf eine kanonische Art und Weise in einen Raum nichtlinearer verallgemeinerter Tensorfelder eingebettet werden können. Dies steht im Gegensatz zu einer verwandten Konstruktion in der an Stelle des Zusammenhanges auf der Mannigfaltigkeit ein zusätzlicher Regularisierungsparameter für verallgemeinerte Tensorfelder eingeführt wurde, was im Vergleich zur vorliegenden Variante technisch aufwändiger ist. Die wesentliche Frage zum konstruierten Raum verallgemeinerter Tensorfelder ist, ob die Einbettung von distributionellen Tensorfeldern mit Pullback entlang von Diffeomorphismen und Lie-Ableitungen kommutiert. Im Allgemeinen ist dies nicht der Fall, was ein Hauptresultat dieser Arbeit darstellt; jedoch erhält man ein positives Ergebnis für solche Operationen, welche die zugrunde liegende Struktur der Riemannschen Mannigfaltigkeit respektieren, das heißt für Pullback entlang von Isometrien beziehungsweise Lie-Ableitungen entlang von Killing-Vektorfeldern. Teil II gibt eine detaillierte Beschreibung der Topologie auf Tensorprodukten von Schnitträumen endlichdimensionaler Vektorbündel, die für die Beschreibung von distributionellen Tensorfeldern nützlich ist. Man erhält dadurch bornologisch isomorphe Darstellungen letzterer als Ergänzung zur vorhandenen Literatur. Teil III schließlich gibt eine Punktwertecharakterisierung für verallgemeinerte Funktionen in der lokalen diffeomorphismeninvarianten Theorie, welche zuvor nur in einfacheren Fällen verfügbar war.This thesis presents three related topics in the field of full diffeomorphism-invariant Colombeau algebras. Part I consists of an extension of the theory of full diffeomorphism-invariant Colombeau algebras to the setting of generalized tensor fields on Riemannian manifolds. The Levi-Civita connection is used as a key element to regularize distributional tensor fields and thus embed them in a canonical way into a space of nonlinear generalized tensor fields. This stands in contrast to a related construction in which instead of a connection on the manifold an additional regularization parameter of generalized tensor fields was used, which is technically more involved. The central question about the constructed space of generalized tensor fields is whether the embedding of distributional tensor fields commutes with pullback along diffeomorphisms and Lie derivatives. In general this is not the case, which is a main result of this work. One gets however a positive answer for operations respecting the structure of the Riemannian manifold, i.e., for pullbacks along isometries and Lie-derivatives along Killing vector fields. Part II gives a detailed description of the topology on tensor products of spaces of sections of finite dimensional vector bundles which is used for the description of distributional tensor fields. One obtains bornologically isomorphic representations of the latter, which complements the existing literature. Part III finally gives a point value characterization for generalized functions in the local full diffeomorphism-invariant theory. Previously, such a characterization has been available only in simpler cases