相关论文: A Baire Category Approach to Besicovitch's Theorem…
We study the uniform computational content of different versions of the Baire Category Theorem in the Weihrauch lattice. The Baire Category Theorem can be seen as a pigeonhole principle that states that a complete (i.e., "large") metric…
Category-measure duality concerns applications of Baire-category methods that have measure-theoretic analogues. The set-theoretic axiom needed in connection with the Baire category theorem is the Axiom of Dependent Choice DC rather than the…
Aim of this paper is to develop a new technique, based on the Baire category theorem, in order to establish the closure of reachable sets and the existence of optimal trajectories for control systems, without the usual convexity…
A Besicovitch set is a subset of $\R^d$ that contains a unit line segment in every direction and the famous Kakeya conjecture states that Besicovitch sets should have full dimension. We provide a number of results in support of this…
The Countable Telescope Conjecture arose in the framework of stable homotopy theory, as a tool conceived to study the chromatic filtration. It turned out, however, to trigger extremely fertile research within the framework of Module…
We prove that the Baire Category Theorem is equivalent to the following: Let G be a topological groupoid such that the unit space is a complete metric space, and there is a countable cover of G by neighbourhood bisections. If G is…
We endow the set of all invariant measures of a topological dynamical system with a metric $\bar{\rho}$, which induces a topology stronger than the the weak$^*$-topology. Then, we study the closedness of ergodic measures within a…
Descriptive set theory is mainly concerned with studying subsets of the space of all countable binary sequences. In this paper we study the generalization where countable is replaced by uncountable. We explore properties of generalized…
The smooth development of large parts of mathematics hinges on the idea that some sets are `small' or `negligible' and can therefore be ignored for a given purpose. The perhaps most famous smallness notion, namely `measure zero', originated…
We prove a model theoretic Baire category theorem for $\tilde\tau_{low}^f$-sets in a countable simple theory in which the extension property is first-order and show some of its applications. We also prove a trichotomy for minimal types in…
The Kestelman-Borwein-Ditor Theorem asserts that a non-negligible subset of $\mathbb{R}$ which is Baire (=has the Baire property, BP) or measurable is shift-compact: it contains some subsequence of any null sequence to within translation by…
We study closed choice principles for different spaces. Given information about what does not constitute a solution, closed choice determines a solution. We show that with closed choice one can characterize several models of…
By using a multiscale analysis, we establish quantitative versions of the Besicovitch projection theorem (almost every projection of a purely unrectifiable set in the plane of finite length has measure zero) and a standard companion result,…
In this paper we study a new approach to classify mathematical theorems according to their computational content. Basically, we are asking the question which theorems can be continuously or computably transferred into each other? For this…
In intuitionistic mathematics, the Brouwer Continuity Theorem states that all total real functions are (uniformly) continuous on the unit interval. We study this theorem and related principles from the point of view of Reverse Mathematics…
We consider an action of a countable amenable group on a compact metric space, focusing on the set of generic points with respect to a fixed F{\o}lner sequence. For a given characteristic class, we prove that the set of points that are…
The Baire category theorem states that every complete pseudometric space is a Baire space. There are some results in metric spaces which have their analogue in uniform spaces, however this is not one of them. Nonetheless, since the Baire…
The paper is a contribution to intuitionistic reverse mathematics. We introduce a formal system called Basic Intuitionistic Mathematics BIM, and then search for statements that are, over BIM, equivalent to Brouwer's Fan Theorem or to its…
This paper continues to study the connection between reverse mathematics and Weihrauch reducibility. In particular, we study the problems formed from Maltsev's theorem on the order types of countable ordered groups. Solomon showed that the…
We examine several classical concepts from topology and functional analysis, using methods of commutative algebra. We show that these various concepts are all controlled by BC R-rings and their maximal spectra. A BC R-ring is a ring A that…