On closed P-sets with ccc in the space omega^*

Reducts of structures and maximal-closed permutation groups

Answering a question of Junker and Ziegler, we construct a countable first order structure which is not omega-categorical, but does not have any proper non-trivial reducts, in either of two senses (model-theoretic, and group-theoretic). We also construct a strongly minimal set which is not omega-categorical but has no proper non-trivial reducts in the model-theoretic sense.

Existentially closed II_1 factors

We examine the properties of existentially closed (R^omega-embeddable) II_1 factors. In particular, we use the fact that every automorphism of an existentially closed (R^omega-embeddable) II_1 factor is approximately inner to prove that Th(R) is not model-complete. We also show that Th(R) is complete for both finite and infinite forcing and use the latter result to prove that there exist continuum many nonisomorphic existentially closed models of Th(R).

How to force a countably tight, initially ω_1-compact and non-compact space?

Improving a result of M. Rabus we force a normal, locally compact, 0-dimensional,Frechet-Uryson, initially omega_1-compact and non-compact space X of size omega_2 having the following property: for every open (or closed) set A in X we have |A|<omega_2 or |X\A|<omega_2.

A hierarchy of maps between compacta

Let $CH$ be the class of compacta (i.e., compact Hausdorff spaces), with $BS$ the subclass of Boolean spaces. For each ordinal $alpha$ and pair $(K,L)$ of subclasses of $CH$, we define $Lev_{>=alpha}(K,L)$, the class of maps of level at least $alpha$ from spaces in $K$ to spaces in $L$, in such a way that, when $alpha < omega$, $Lev}_{>=alpha}(BS,BS)$ consists of the Stone duals of Boolean lattice embeddings that preserve all prenex first-order formulas of quantifier rank $alpha$. Maps of level $>=0$ are just the continuous surjections, and the maps of level $>=1$ are the co-existential maps. …

Baire Category for Monotone Sets

We study Baire category for subsets of 2^omega that are downward-closed with respect to the almost-inclusion ordering (on the power set of the natural numbers, identified with 2^omega). We show that it behaves better in this context than for general subsets of 2^omega. In the downward-closed context, the ideal of meager sets is prime and b-complete (where b is the bounding number), while the complementary filter is g-complete (where g is the groupwise density cardinal). We also discuss other cardinal characteristics of this ideal and this filter, and we show that analogous results for measure …

A friendly iteration forcing that the four cardinal characteristics of $\mathcal{E}$ can be pairwise different

Let $\mathcal{E}$ be the $σ$-ideal generated by the closed measure zero sets of reals. We use an ultrafilter-extendable matrix iteration of ccc posets to force that, for $\mathcal{E}$, their associated cardinal characteristics (i.e.\ additivity, covering, uniformity and cofinality) are pairwise different.

一个重复点给出 {ω_1} 上的闭无界超滤子

One Repeat Point Gives a Closed, Unbounded Ultrafilter on {ω_1}

本文确定关于 ω_1 的闭无界超滤子成为真正超滤子所需的一致性强度恰为 ZFC 加一个可测基数。

Forcing tightness in products of fans

The theta--fan F_theta is the quotient space obtained by identifying the non--isolated points of the product theta \times (omega+1) with a single point \infty. (Here, theta has the discrete topology and omega+1 has the order topology.) We define several notions of forcing that allow us to manipulate the tightness t of products of fans. Some consequences include: t (F_theta \times F_omega) = theta does not imply the existence of a (theta,omega)--gap, new examples of first countable <theta--collectionwise Hausdorff (cwH) spaces that are not \leq theta--cwH for singular cardinals theta, and for c…

Anti-Urysohn spaces

All spaces are assumed to be infinite Hausdorff spaces. We call a space "anti-Urysohn" $($AU in short$)$ iff any two non-emty regular closed sets in it intersect. We prove that $\bullet$ for every infinite cardinal $κ$ there is a space of size $κ$ in which fewer than $cf(κ)$ many non-empty regular closed sets always intersect; $\bullet$ there is a locally countable AU space of size $κ$ iff $ω\le κ\le 2^{\mathfrak c}$. A space with at least two non-isolated points is called "strongly anti-Urysohn" $($SAU in short$)$ iff any two infinite closed sets in it intersect. We prove that $\bullet$ if $X…

Automorphisms and strongly invariant relations

We investigate characterizations of the Galois connection sInv-Aut between sets of finitary relations on a base set A and their automorphisms. In particular, for A=omega_1, we construct a countable set R of relations that is closed under all invariant operations on relations and under arbitray intersections, but is not closed under sInv(Aut(-)). Our structure (A,R) has an omega-categorical first order theory. A higher order definable well-order makes it rigid, but any reduct to a finite language is homogeneous.

居间关系与$D$-集的Omega-范畴极限

Omega-categorical limits of betweenness relations and $D$-sets

本文研究两类寡形Jordan置换群构造，解决了相关开放问题，证明$D$-关系极限可齐性化，明确其NIP性质、轨道增长速率等特征，并验证其自同构群为对称群中的极大闭子群。