Distributive proper forcing axiom and a left-right dichotomy of Cichon's diagram

In this paper, we study distributive proper forcing axiom(DPFA) and prove its consistency with a dichotomy of the Cichon's diagram, relative to certain large cardinal assumption. Namely, we evaluate the cardinal invariants in Cichon's diagram with the first two uncountable cardinals in the way that the left-hand side has the least possible cardinality while the right-hand side has the largest possible value, and preserve the evaluation along the way of forcing DPFA.     

On the consistency strength of the proper forcing axiom

In recent work, the second author extended combinatorial principles due to Jech and Magidor that characterize certain large cardinal properties so that they can also hold true for small cardinals. For inaccessible cardinals, these modifications have no effect, and the resulting principles still give the same characterization of large cardinals. We prove that the proper forcing axiom PFA implies these principles hold for ω₂. Using this, we argue to show that any of the known methods for forcing models of PFA from a large cardinal assumption requires a strongly compact cardinal. If one forces PFA using a proper forcing, then we get the optimal result that a supercompact cardinal is necessary.     

Many simple cardinal invariants

Summary Forg in we definec(f,g) be the least number of uniform trees withg-splitting needed to cover a uniform tree withf-splitting. We show that we can simultaneously force ₁ many different values for different functions (f,g). In the language of [B1]: There may be ₁ many distinct uniformII ₁ ¹ characteristics.Supported by Israeli Academy of Sciences, Basic Research FundPublication 448. Supported partially by Israeli Academy of Sciences, Basic Research Fund and by the Edmund Landau Center for research in Mathematical Analysis (supported by the Minerva Foundation, Germany)     

Sacks forcing,Laver forcing,and Martin's axiom

Summary In this paper we study the question assuming MA+CH does Sacks forcing or Laver forcing collapse cardinals? We show that this question is equivalent to the question of what is the additivity of Marczewski's ideals ⁰. We give a proof that it is consistent that Sacks forcing collapses cardinals. On the other hand we show that Laver forcing does not collapse cardinals.Research partially supported by NSF grant 8801139     

Proper forcing axiom and selective separability

We continue the study of Selectively Separable (SS) and, a game-theoretic strengthening, strategically selectively separable spaces (SS⁺) (see Barman, Dow (2011) [1]). The motivation for studying SS⁺ is that it is a property possessed by all separable subsets of Cp(X) for each σ-compact space X. We prove that the winning strategy for countable SS⁺ spaces can be chosen to be Markov. We introduce the notion of being compactlike for a collection of open sets in a topological space and with the help of this notion we prove that there are two countable SS⁺ spaces such that the union fails to be SS⁺, which contrasts the known result about SS spaces. We also prove that the product of two countable SS⁺ spaces is again countable SS⁺. One of the main results in this paper is that the proper forcing axiom, PFA, implies that the product of two countable Fréchet spaces is SS, a statement that was shown in Barman, Dow (2011) [1] to consistently fail. An auxiliary result is that it is consistent with the negation of CH that all separable Fréchet spaces have π-weight at most ω₁.     

Even more simple cardinal invariants

Using GCH, we force the following: There are continuum many simple cardinal characteristics with pairwise different values. Supported by a European Union Marie Curie EIF fellowship, contract MEIF-CT-2006-024483.     

A survey of cardinal invariants

This expository paper describes a number of theorems dealing with cardinal numbers associated with topoligical spaces. The weight, density character, and cellular number receive chief attention. A few of the theorems cited-e.g., the Hewitt-Pondionery-Marczewski theorem on the density character of a product space, and the Erdös-Tarski theorem asserting that the cellular number, unless uncountable and weakly inaccessible, is achieved — has been known for many years; proofs are included. Other equally exciting results of more recent vintage, such as Arhangel'skii's solution to the Alexandroff problem, the Hajnal-Jokise evaluation of the number of sets in a Hausdorff topology, and Nyiko's solution to the Engelking-Mröwka problem, have perhaps not yet achieved their optimal statement and/or proof, these results are stated here as carefully and completely as time and the present author's ability permit, but the proofs are only sketched, or omitted entirely. The following also are briefly mentioned: Smirnov's theorem w(ßX) = C¹(X); the F.K. Schmidt A.H. Stone-Kruse theorem |C(X)|=|C(X)|^ℵ₀; the relation m^l = m in Stonian spaces; the Noble-Ulmer cluster of theorems on preservation of certain cardinal properties under passage to products; and Mrówka's theorems on embedding the discrete space D(m) as a closed subspace of N^m.     

Some cardinal invariants on the space

Let C_α(X,Y) be the set of all continuous functions from X to Y endowed with the set-open topology where α is a hereditarily closed, compact network on X such that closed under finite unions. We define two properties (E1) and (E2) on the triple (α,X,Y) which yield new equalities and inequalities between some cardinal invariants on C_α(X,Y) and some cardinal invariants on the spaces X, Y such as: Theorem If Y is an equiconnected space with a base consisting of φ-convex sets, then for each fC(X,Y), χ(f,C_α(X,Y))=αa(X).w_e(f(X)).Corollary Let Y be a noncompact metric space and let the triple (α,X,Y) satisfy (E1). The following are equivalent:

(i) C_α(X,Y) is a first-countable space.

(ii) π-character of the space C_α(X,Y) is countable.

(iii) C_α(X,Y) is of pointwise countable type.

(iv) There exists a compact subset K of C_α(X,Y) such that π-character of K in the space C_α(X,Y) is countable.

(v) αa(X)₀.

(vi) C_α(X,Y) is metrizable.

(vii) C_α(X,Y) is a q-space.

(viii) There exists a sequence of nonempty open subset of C_α(X,Y) such that each sequence with g_nO_n for each nω, has a cluster point in C_α(X,Y).

Constructing Boolean Algebras for cardinal invariants

We construct Boolean Algebras answering some questions of J. Donald Monk on cardinal invariants. The results are proved in ZFC (rather than giving consistency results). We deal with the existence of superatomic Boolean Algebras with 'few automorphisms', with entangled sequences of linear orders, and with semi-ZFC examples of the non-attainment of the spread (and hL, hd).     

On cardinal invariants in some classes of topological spaces

Cardinal invariants of some topological linearly ordered spaces and of topological spaces which are representable as the union of a linearly ordered (by inclusion) family of compact spaces are investigated. As one of the results, it is proved that if a paracompact space T is representable as the union of a chain of compact subspaces, then T is a Lindelöf space.     

Inner models with large cardinal features usually obtained by forcing

We construct a variety of inner models exhibiting features usually obtained by forcing over universes with large cardinals. For example, if there is a supercompact cardinal, then there is an inner model with a Laver indestructible supercompact cardinal. If there is a supercompact cardinal, then there is an inner model with a supercompact cardinal κ for which 2 ^κ ?=?κ ⁺, another for which 2 ^κ ?=?κ ⁺⁺ and another in which the least strongly compact cardinal is supercompact. If there is a strongly compact cardinal, then there is an inner model with a strongly compact cardinal, for which the measurable cardinals are bounded below it and another inner model W with a strongly compact cardinal κ, such that ${H^{V}_{\kappa^+} \subseteq {\rm HOD}^W}$ . Similar facts hold for supercompact, measurable and strongly Ramsey cardinals. If a cardinal is supercompact up to a weakly iterable cardinal, then there is an inner model of the Proper Forcing Axiom and another inner model with a supercompact cardinal in which GCH?+?V?=?HOD holds. Under the same hypothesis, there is an inner model with level by level equivalence between strong compactness and supercompactness, and indeed, another in which there is level by level inequivalence between strong compactness and supercompactness. If a cardinal is strongly compact up to a weakly iterable cardinal, then there is an inner model in which the least measurable cardinal is strongly compact. If there is a weakly iterable limit δ of <δ-supercompact cardinals, then there is an inner model with a proper class of Laver-indestructible supercompact cardinals. We describe three general proof methods, which can be used to prove many similar results.     

Products of regular cardinals and cardinal invariants of products of Boolean algebras

We answer some questions of Monk, and give some information on others concerning cardinal invariants of Boolean algebras under ultraproducts and products. The author would like to thank the United States-Israel Binational Science Foundation for partially supporting this research, and Alice Leonhardt for the beautful typing. Publication no. 345. §1–4 of this paper are essentially the letters which the author sent in December 1987 to Monk solving problems from his notes on cardinal invariants of B.A.; §8 and §9 were written for §4; the other sections, §§5, 6 and 7, were completed in March 1988. Concerning §§5–9, for further results see Abstracts of AMS and subsequent papers. §10 was written during the Arcata meeting, summer 1985, and §11 in January 1986, after questions of Todorcevic.     

First countability, tightness, and other cardinal invariants in remainders of topological groups

We consider the following natural questions: when a topological group G has a first countable remainder, when G has a remainder of countable tightness? This leads to some further questions on the properties of remainders of topological groups. Let G be a topological group. The following facts are established. 1. If Gω has a first countable remainder, then either G is metrizable, or G is locally compact. 2. If G has a countable network and a first countable remainder, then either G is separable and metrizable, or G is σ-compact. 3. Under (MA+¬CH) every topological group with a countable network and a first countable remainder is separable and metrizable. Some new open problems are formulated.     

On the structure invariants of proper rational matrices with prescribed finite poles

Abstract

The algebraic structure of matrices defined over arbitrary fields whose elements are rational functions with no poles at infinity and prescribed finite poles is studied. Under certain very general conditions, they are shown to be matrices over an Euclidean domain that can be classified according to the corresponding invariant factors. The relationship between these invariants and the local Wiener–Hopf factorization indices will be clarified. This result can be seen as an extension of the classical theorem on pole placement by Rosenbrock in control theory.