Related papers: There may be no Hausdorff ultrafilters
We prove that the existence of a selective ultrafilter implies the existence of a countably compact Hausdorff group topology on the free Abelian group of size continuum. As a consequence, we show that the existence of a selective…
In this paper we present some bounds of Hausdorff measures of objects definable in o-minimal structures: sets, fibers of maps, inverse images of curves of maps, etc. Moreover, we also give some explicit bounds for semi-algebraic or…
We give a short proof that any non-zero Euclidean space has a compact subset of Hausdorff dimension one that contains a differentiability point of every real-valued Lipschitz function defined on the space.
Cantor sets in \(\mathbb{R}\) are common examples of sets for which Hausdorff measures can be positive and finite. However, there exist Cantor sets for which no Hausdorff measure is supported and finite. The purpose of this paper is to try…
We further investigate a divisibility relation on the set $\beta N$ of ultrafilters on the set of natural numbers. We single out prime ultrafilters (divisible only by 1 and themselves) and establish a hierarchy in which a position of every…
We prove that \textsf{P}-points (even strong P-points) and Gruff ultrafilters exist in any forcing extension obtained by adding fewer than $\aleph_{\omega}% $-many random reals to a model of \textsf{CH. }These results improve and correct…
We show that, consistently, there is an ultrafilter F on omega such that if N^l_n=(P^l_n cup Q^l_n,P^l_n,Q^l_n,R^l_n) (for l =1,2, n< omega), P^l_n cup Q^l_n subseteq omega, and prod_{n<omega} N^1_n/F and prod_{n< omega}N^2_n/F are…
We exhibit a forcing for producing a model with no nowhere dense ultrafilters that satisfies the full Sacks Property. By interleaving this forcing with other forcing notions, a model containing a $(2, {\aleph}_{0})$-selective ultrafilter,…
We show that it is not provable in ZFC that any two countable elementarily equivalent structures have isomorphic ultrapowers relative to some ultrafilter on omega .
A set of natural numbers will be called \emph{harmonic} if the reciprocals of its elements form a divergent series. An ultrafilter of the natural numbers will he called \emph{harmonic} if all each members are harmonic sets. The harmonic…
The ultrafilters on the partial order $([\omega]^{\omega},\subseteq^*)$ are the free ultrafilters on $\omega$, which constitute the space $\omega^*$, the Stone-Cech remainder of $\omega$. If $U$ is an upperset of this partial order (i.e., a…
Under MA we prove that for the ideal $\cal I$ of thin sets on $\omega$ and for any ordinal $\gamma \leq \omega_1$ there is an ${\cal I}$-ultrafilter (in the sense of Baumgartner), which belongs to the class ${\cal P}_{\gamma}$ of…
We are interested in generalizing part of the theory of ultrafilters on omega to larger cardinals. Here we set the scene for further investigations introducing properties of ultrafilters in strong sense dual to being normal.
In the absence of the Axiom of Choice, necessary and sufficient conditions for a locally compact Hausdorff space to have all non-empty second-countable compact Hausdorff spaces as remainders are given in $\mathbf{ZF}$. Among other…
The \emph{Filter Dichotomy} says that every uniform nonmeager filter on the integers is mapped by a finite-to-one function to an ultrafilter. The consistency of this principle was proved by Blass and Laflamme. A function between topological…
We investigate mutual behavior of cascades, contours of which are contained in a fixed ultrafilter. Using that relation we prove (ZFC) that the class of strict $J_{\omega^\omega}$-ultrafilters, introduced by J. E. Baumgartner in…
Let $U$ be an absolute ultrafilter on the set of non-negative integers $\mathbb{N}$. For any sequence $x=(x_n)_{n\geq 0}$ of real numbers, let $U(x)$ denote the topological filter consisting of the open sets $W$ of $\mathbb{R}$ with $\{n…
We establish a formula yielding the Hausdorff measure for a class of non-self-similar Cantor sets in terms of the canonical covers of the Cantor set.
Even a functor without an adjoint induces a monad, namely, its codensity monad; this is subject only to the existence of certain limits. We clarify the sense in which codensity monads act as substitutes for monads induced by adjunctions. We…
A countably based profinite group can be naturally seen as a metric space with respect to a given filtration, and thus, it has a well defined Hausdorff dimension function. Barnea and Shalev found a group theoretical expression for the…