On the hereditary paracompactness of locally compact, hereditarily normal spaces

We establish that if it is consistent that there is a supercompact cardinal, then it is consistent that every locally compact, hereditarily normal space which does not include a perfect pre-image of omega_1 is hereditarily paracompact

PFA(S)[S] and the Arhangel'skii-Tall problem

We discuss the Arhangel'skii-Tall problem and related questions in models obtained by forcing with a coherent Souslin tree

Optimal Matrices of Partitions and an Application to Souslin Trees

The basic result of this note is a statement about the existence of families of partitions of the set of natural numbers with some favourable properties, the n-optimal matrices of partitions. We use this to improve a decomposition result for strongly homogeneous Souslin trees. The latter is in turn applied to separate strong notions of rigidity of Souslin trees, thereby answering a considerable portion of a question of Fuchs and Hamkins.Comment: 19 pages, submitted to Fundamenta Mathematica

PFA(S)[S] for the masses

We present S. Todorcevic's method of forcing with a coherent Souslin tree over restricted iteration axioms as a black box usable by those who wish to avoid its complexities but still access its power

Chain homogeneous Souslin algebras

Assuming Jensen's principle diamond-plus we construct Souslin algebras all of whose maximal chains are pairwise isomorphic as total orders, thereby answering questions of Koppelberg and TodorcevicComment: 29 pages, submitted to the Mathematical Logic Quarterl

Normality versus paracompactness in locally compact spaces

This note provides a correct proof of the result claimed by the second author that locally compact normal spaces are collectionwise Hausdorff in certain models obtained by forcing with a coherent Souslin tree. A novel feature of the proof is the use of saturation of the non-stationary ideal on \omega_1, as well as of a strong form of Chang's Conjecture. Together with other improvements, this enables the characterization of locally compact hereditarily paracompact spaces as those locally compact, hereditarily normal spaces that do not include a copy of \omega_1

Souslin Algebra Embeddings

A Souslin algebra is a complete Boolean algebra whose main features are ruled by a tight combination of an antichain condition with an infinite distributive law. The present article divides into two parts. In the first part a representation theory for the complete and atomless subalgebras of Souslin algebras is established (building on ideas of Jech and Jensen). With this we obtain some basic results on the possible types of subalgebras and their interrelation. The second part begins with a review of some generalizations of results from descriptive set theory concerning Baire category which are then used in non-trivial Souslin tree constructions that yield Souslin algebras with a remarkable subalgebra structure. In particular, we use this method to prove that under the diamond principle there is a bi-embeddable though not isomorphic pair of homogeneous Souslin algebras.Comment: 41 pages, 2 figures; submitted to: Archive for Mathematical Logi

PFA(S)[S] and Locally Compact Normal Spaces

We examine locally compact normal spaces in models of form PFA(S)[S], in particular characterizing paracompact, countably tight ones as those which include no perfect pre-image of omega_1 and in which all separable closed subspaces are Lindelof

A Microscopic approach to Souslin-tree constructions. Part I

We propose a parameterized proxy principle from which $\kappa$-Souslin trees with various additional features can be constructed, regardless of the identity of $\kappa$. We then introduce the microscopic approach, which is a simple method for deriving trees from instances of the proxy principle. As a demonstration, we give a construction of a coherent $\kappa$-Souslin tree that applies also for $\kappa$ inaccessible. We then carry out a systematic study of the consistency of instances of the proxy principle, distinguished by the vector of parameters serving as its input. Among other things, it will be shown that all known $\diamondsuit$-based constructions of $\kappa$-Souslin trees may be redirected through this new proxy principle.Comment: 43 page

Hereditarily normal manifolds of dimension > 1 may all be metrizable

P.J. Nyikos has asked whether it is consistent that every hereditarily normal manifold of dimension > 1 is metrizable, and proved it is if one assumes the consistency of a supercompact cardinal, and, in addition, that the manifold is hereditarily collectionwise Hausdorff. We are able to omit these extra assumptions