Related papers: All meager filters may be null
A topological space X$ has the Frechet-Urysohn property if for each subset A of X and each element x in the closure of A, there exists a countable sequence of elements of A which converges to x. Reznichenko introduced a natural…
We investigate the properties of ideals associated with Kuratowski partitions of non-complete Baire metric spaces. We show that such an ideal can be precipitous.
Let dec be the least cardinal kappa such that every function of first Baire class can be decomposed into kappa continuous functions. Cichon, Morayne, Pawlikowski and Solecki proved that cov(Meager) <= dec <= d and asked whether these…
For almost all Riemannian metrics (in the $C^\infty$ Baire sense) on a compact manifold with boundary $(M^{n+1},\partial M)$, $3\leq (n + 1)\leq 7$, we prove that, for any open subset $V$ of $\partial M$, there exists a compact, properly…
A compact space $X$ is called $\pi$-monolithic if for any surjective continuous mapping $f:X\rightarrow K$ where $K$ is a metrizable compact space there exists a metrizable compact space $T\subseteq X$ such that $f(T)=K$. A topological…
In 1985 S.~Saeki and K.~Stromberg published the following question: {\it Does every infinite compact group have a subgroup which is not Haar measurable?} An affirmative answer is given for all compact groups with the exception of some…
We show that, assuming the existence of $\mathfrak{c}$ incomparable selective ultrafilters, there exists a Wallace semigroup whose infinite countable power is the least power which fails to be countably compact. This answers positively…
A result of H. Steinhaus states that any positive Lebesgue measurable set has a property that its difference set contains an open interval around the origin. Y. V. Mospan proved that this result is the characterization of absolutely…
We present a useful technique of proving strong $\mathfrak{c}$-algebrability. As an outcome we obtain the strong $\mathfrak{c}$-algebrability of the following sets of functions: strong Sierpi\'nski-Zygmund, nowhere H\"older, Bruckner-Garg,…
Rademacher theorem states that every Lipschitz function on the Euclidean space is differentiable almost everywhere, where "almost everywhere" refers to the Lebesgue measure. In this paper we prove a differentiability result of similar type,…
In this paper we shall introduce some nonmeasurable and completely nonmeasurable subsets of the plane with various additional properties, e.g. being Hamel basis, intersecting each line in a strong Luzin / Sierpi\'nski set. Also some…
The evoluted set is the set of configurations reached from an initial set via a fixed flow for all times in a fixed interval. We find conditions on the initial set and on the flow ensuring that the evoluted set has negligible boundary (i.e.…
We prove the following two results. 1. If $X$ is a completely regular space such that for every topological space $Y$ each separately continuous function $f:X\times Y\to\mathbb R$ is of the first Baire class, then every Lindel\"of subspace…
We characterize the Hurewicz covering property in metrizable spaces in terms of properties of the metrics of the space. Then we show that a weak version of selective screenability, when combined with the Hurewicz property, implies selective…
We give a short proof of an improved version of the Effros Open Mapping Principle via a shift-compactness theorem (also with a short proof), involving `sequential analysis' rather than separability, deducing it from the Baire property in a…
We prove that if $G$ is a totally bounded abelian group \st\ its dual group $\widehat{G}_p$ equipped with the finite-open topology is a Baire group, then every compact subset of $G$ must be finite. This solves an open question by Chasco,…
We prove theorems of the following form: if $A\subseteq {\mathbb R}^2$ is a big set, then there exists a big set $P\subseteq {\mathbb R}$ and a perfect set $Q\subseteq {\mathbb R}$ such that $P\times Q\subseteq A$. We discuss cases where…
We show that for a $\sigma $-ideal $\ci$ with a Steinhaus property defined on Banach space, if two non-homeomorphic Banach with the same cardinality of the Hamel basis then there is a $\ci$ nonmeasurable subset as image by any isomorphism…
We show that $ZF+DC+$"all Turing invariant sets of reals have the perfect set property" implies that all sets of reals have the perfect set property. We also show that this result generalizes to all countable analytic equivalence relations.
Work in the measure algebra of the Lebesgue measure on the Cantor space: for comeager many $[A]$ the set of points $x$ such that the density of $x $ at $A$ is not defined is $\Sigma^{0}_{3}$-complete; for some compact $K$ the set of points…