Related papers: Covering of the null ideal may have countable cofi…

Let M be a 3-manifold (possibly with boundary). We show that, for any positive integer g, there exists an open nonempty set of metrics on M for each of which there are stable compact embedded minimal surfaces of genus g with arbitrarily…

A commutative ring R is said to be coverable if it is the union of its proper subrings and said to be finitely coverable if it is the union of a finite number of them. In the latter case, we denote by {\sigma}(R) the minimal number of…

Let R denote a commutative Noetherian ring and let I be an ideal of R such that H_i^I(R) = 0, for all integers i greater than or equal to 2. In this paper we shall prove some results concerning the homological properties of I.

We demonstrate the simple and deep equivalence between quantum coherence and nonclassicality and the definite way in which they determine metrological resolution. Moreover, we define a coherence observable consistent with a classical…

This article is concerned with measure equivalence and uniform measure equivalence of locally compact, second countable groups. We show that two unimodular, locally compact, second countable groups are measure equivalent if and only if they…

A subset of the Cantor cube is null-additive if its algebraic sum with any null set is null. We construct a set of cardinality continuum such that: all continuous images of the set into the Cantor cube are null-additive, it contains a…

We prove a version of $Z$-set unknotting theorem for uncountable products of real numbers.

We prove that the Lipschitz-free space over a countable compact metric space is isometric to a dual space and has the metric approximation property.

We prove the interior and boundary null-controllability of some parabolic evolutions with controls acting over measurable sets.

We continue our study of ends non-compact manifolds. The over-arching aim is to provide an appropriate generalization of Siebenmann's famous collaring theorem that applies to manifolds having non-stable fundamental group systems at…

Suppose that we have a finite colouring of the reals. What sumset-type structures can we hope to find in some colour class? One of our aims is to show that there is such a colouring for which no uncountable set has all of its pairwise sums…

We investigate conditions under which a co-computably enumerable closed set in a computable metric space is computable and prove that in each locally computable computable metric space each co-computably enumerable compact manifold with…

In this paper we examine the commutativity of ideal extensions. We introduce methods of constructing such extensions, in particular we construct a noncommutative ring T which contains a central and idempotent ideal I such that T/I is a…

Let $\mathcal{N}$ be the $\sigma$-ideal of the null sets of reals. We introduce a new property of forcing notions that enable control of the additivity of $\mathcal{N}$ after finite support iterations. This is applied to answer some open…

Let T be the family of open subsets of a topological space (not necessarily Hausdorff or even T_0). We prove that if T has a countable base and is not countable, then T has cardinality at least continuum.

Let $T$ be a countable complete first-order theory with a definable, infinite, discrete linear order. We prove that $T$ has continuum-many countable models. The proof is purely first-order, but raises the question of Borel completeness of…

We study a measure-theoretic notion of connectedness for sets of finite perimeter in the setting of doubling metric measure spaces supporting a weak $(1,1)$-Poincar\'{e} inequality. The two main results we obtain are a decomposition theorem…

We prove that in a Euclidean space of dimension at least two, there exists a compact set of Lebesgue measure zero such that any real-valued Lipschitz function defined on the space is differentiable at some point in the set. Such a set is…

Using a countable support product of creature forcing posets, we show that consistently, for uncountably many different functions the associated Yorioka ideals' uniformity numbers can be pairwise different. In addition we show that, in the…

In this paper we establish that the functor of idempotent probability measures acting in the category of compacta and their continuous mappings is perfect metrizable.