Related papers: Covering by discrete and closed discrete sets

We show that it is consistent relative to a huge cardinal that for all infinite cardinals $\kappa$, $\square_\kappa$ holds and there is a stationary $S \subseteq \kappa^+$ such that $\mathrm{NS}_{\kappa^+} \restriction S$ is…

Given a completely metrizable space $X$, let $\mathfrak{par}(X)$ denote the smallest possible size of a partition of $X$ into Polish spaces, and $\mathfrak{cov}(X)$ the smallest possible size of a covering of $X$ with Polish spaces. Observe…

In our paper [18] we showed that a Tychonoff space $X$ is a $\Delta$-space (in the sense of [20], [30]) if and only if the locally convex space $C_{p}(X)$ is distinguished. Continuing this research, we investigate whether the class $\Delta$…

A famous result of Klee from 1981 is that the Banach space $\ell_1(\kappa)$ admits a disjoint tiling by balls of radius $1$, for all cardinals $\kappa$ with $\kappa^\omega =\kappa$. Klee also observed that the smallest cardinal in which…

Let (X,d) be a metric space and (\Omega, d) a compact subspace of X which supports a non-atomic finite measure m. We consider `natural' classes of badly approximable subsets of \Omega. Loosely speaking, these consist of points in \Omega…

Given a closed Riemannian manifold $(N^{n+1},g)$, $n+1 \geq 3$ we prove the compactness of the space of singular, minimal hypersurfaces in $N$ whose volumes are uniformly bounded from above and the $p$-th Jacobi eigenvalue $\lambda_p$'s are…

We consider special subclasses of the class of Lindel\"of Sigma-spaces obtained by imposing restrictions on the weight of the elements of compact covers that admit countable networks: A space $X$ is in the class $L\Sigma(\leq\kappa)$ if it…

We prove the consistency of: for suitable strongly inaccessible cardinal lambda the dominating number, i.e. the cofinality of lambda^lambda is strictly bigger than cov(meagre_lambda), i.e. the minimal number of nowhere dense subsets of…

Our main result is that, given a collection $\mathcal{R}$ of meager relations on a Polish space $X$ such that $|\mathcal{R}|\leq\omega$, there exists a dense Baire subspace $F$ of $X$ (equivalently, a nowhere meager subset $F$ of $X$) such…

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…

We study the influence of the existence of large cardinals on the existence of wellorderings of power sets of infinite cardinals $\kappa$ with the property that the collection of all initial segments of the wellordering is definable by a…

We prove two $\mathrm{ZFC}$ inequalities between cardinal invariants. The first inequality involves cardinal invariants associated with an analytic P-ideal, in particular the ideal of subsets of $\omega$ of asymptotic density $0$. We obtain…

We mainly investigate model of set theory with restricted choice, e.g., ZF + DC + "the family of countable subsets of lambda is well ordered for every lambda" (really local version for a given lambda). In this frame much of pcf theory can…

In this paper we devote to spaces that are not homotopically hausdorff and study their covering spaces. We introduce the notion of small covering and prove that every small covering of $X$ is the universal covering in categorical sense.…

Starting from a model with a Laver-indestructible supercompact cardinal $\kappa$, we construct a model of $ZF+DC_{\kappa}$ where there are no $\kappa$-mad families.

Definable stationary sets, and specifically, ordinal definable ones, play a significant role in the study of canonical inner models of set theory and the class HOD of hereditarily ordinal definable sets. Fixing a certain notion of…

Assuming the existence of a strong cardinal, we find a model of ZFC in which for each uncountable regular cardinal $\lambda,$ there is no universal graph of size $\lambda$.

In this paper for each cardinal $\kappa$ we construct an infinite $\kappa$-bounded (and hence countably compact) regular space $R_{\kappa}$ such that for any $T_1$ space $Y$ of pseudo-character $\leq\kappa$, each continuous function…

Let $f$ be a generically finite polynomial map $f: \mathbb{C}^n\to \mathbb{C}^m$ of algebraic degree $d$. Motivated by the study of the Jacobian Conjecture, we prove that the set $S_f$ of non-properness of $f$ is covered by parametric…

Let $X$ be a locally compact topological space, $(Y,d)$ be a boundedly compact metric space and $LB(X,Y)$ be the space of all locally bounded functions from $X$ to $Y$. We characterize compact sets in $LB(X,Y)$ equipped with the topology of…