Related papers: Point-cofinite covers in the Laver model

Considering the space of closed subsets of $\mathbb{R}^d$, endowed with the Chabauty-Fell topology, and the affine action of $SL_d(\mathbb{R})\ltimes\mathbb{R}^d$, we prove that the only minimal subsystems are the fixed points…

We study representations of a Leavitt path algebra $L$ of a finitely separated digraph $\Gamma$ over a field. We show that the category of $L$-modules is equivalent to a full subcategory of quiver representations. When $\Gamma$ is a…

Let $\kappa$ be an infinite regular cardinal. We define a topological space $X$ to be $T_{\kappa-Borel}$-space (resp. a $T_{\kappa-BP}$-space) if for every $x\in X$ the singleton $\{x\}$ belongs to the smallest $\kappa$-additive algebra of…

We generalize the basic theory of universally Baire sets of $2^\omega$ to a theory of universally Baire subsets of $2^\kappa$. We show that the fundamental characterizations of the property of being universally Baire have natural…

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 base of cardinality <= mu, lambda <= mu < 2^lambda, lambda strong limit of cofinality aleph_0, then T has…

In their paper titled "On $F$-inverse covers of inverse monoids", Auinger and Szendrei have shown that every finite inverse monoid has an $F$-inverse cover if and only if each finite graph admits a locally finite group variety with a…

For a group G with trivial center there is a natural embedding of G into its automorphism group, so we can look at the latter as an extension of the group. So an increasing continuous sequence of groups, the automorphism tower, is defined,…

We study productive properties of gamma spaces, and their relation to other, classic and modern, selective covering properties. Among other things, we prove the following results: 1. Solving a problem of F. Jordan, we show that for every…

In this paper we demonstrate that it is consistent, relative to the existence of a supercompact cardinal, that there is no linear order which is minimal with respect to being non $\sigma$-scattered. This shows that a theorem of Laver, which…

We prove the following two results. Theorem A: Let alpha be a limit ordinal. Suppose that 2^{|alpha|}<aleph_alpha and 2^{|alpha|^+}<aleph_{|alpha|^+}, whereas aleph_alpha^{|alpha|}>aleph_{|alpha|^+}. Then for all n< omega and for all…

In this paper a hypergraph will be identified with the family of its edges. A hypergraph $\mathcal E$ possesses property $C(k,{\rho})$ iff $|\bigcap \mathcal E'|<{\rho}$ for each $\mathcal E'\in {[\mathcal E]}^{k}$. A vertex set $Y\subset…

We define the compact universal cover of a compact, metrizable connected space (i.e. a continuum) X to be the inverse limit of all continua that regularly cover X. We show that such covers do indeed form an inverse system with bonding maps…

We analyze the notion of guessing model, a way to assign combinatorial properties to arbitrary regular cardinals. Guessing models can be used, in combination with inaccessibility, to characterize various large cardinals axioms, ranging from…

Various problems on integers lead to the class of congruence preserving functions on rings, i.e. functions verifying $a-b$ divides $f(a)-f(b)$ for all $a,b$. We characterized these classes of functions in terms of sums of rational…

We study star-finite coverings of infinite-dimensional normed spaces. A family of sets is called star-finite if each of its members intersects only finitely many other members of the family. It follows by our results that an LUR or a…

We show that the statement "every universally Baire set of reals has the perfect set property" is equiconsistent modulo ZFC with the existence of a cardinal that we call a virtually Shelah cardinal. These cardinals resemble Shelah cardinals…

Kunen's proof of the non-existence of Reinhardt cardinals opened up the research on very large cardinals, i.e., hypotheses at the limit of inconsistency. One of these large cardinals, I0, proved to have descriptive-set-theoretical…

In this article we prove three main theorems: (1) guessing models are internally unbounded, (2) for any regular cardinal $\kappa \ge \omega_2$, $\textsf{ISP}(\kappa)$ implies that $\textsf{SCH}$ holds above $\kappa$, and (3) forcing posets…

Results of Sierpinski and others have shown that certain finite-dimensional product sets can be written as unions of subsets, each of which is "narrow" in a corresponding direction; that is, each line in that direction intersects the subset…

A 1984 problem of S.Z. Ditor asks whether there exists a lattice of cardinality aleph two, with zero, in which every principal ideal is finite and every element has at most three lower covers. We prove that the existence of such a lattice…