Martin's Axiom Research Articles

Ladder systems on trees

We formulate the notion of uniformization of colorings of ladder systems on subsets of trees. We prove that Suslin trees have this property and also Aronszajn trees in the presence of Martin's Axiom. As an application we show that if a tree has this property, then every countable discrete family of subsets of the tree can be separated by a family of pairwise disjoint open sets. Such trees are then normal and hence countably paracompact. As a dual result for special Aronszajn trees we prove that the weak diamond, Φ ω , implies that no special Aronszajn tree can be countably paracompact.

Weakly Ramsey Sets in Banach Spaces

Gowers' analysis of the combinatorial content of his celebrated dichotomy for infinite-dimensional separable Banach spaces [7] led him to the formulation of the property of being weakly Ramsey applied to sets of block bases, a combinatorial notion related to the classical Ramsey property for infinite sets of positive integers. Let [N] be the set of all infinite sets of positive integers. With the natural topology induced by the Cantor space via characteristic functions, [N] is a Polish space. A subset _ [N] is Ramsey if there exists A # [N] such that either [A] _ or [A] & _=<, where [A] is the set of all infinite subsets of A. The famous Galvin Prikry theorem asserts that every Borel subset of [N] is Ramsey ([4]) or, equivalently, that every Borel map from [N] into a finite space is constant on a cube [A]. Silver [17] shows that, in fact, all analytic subsets of [N], i.e., the continuous images of Borel sets, are Ramsey. A simpler, more combinatorial proof is given by Ellentuck [3]. The Ramsey property for more complex subsets of [N] turns out to depend essentially on the axioms of set theory. Thus, for instance, while Go del's axiom of constructibility implies that some continuous image of a co-analytic set is not Ramsey (see [9]), Martin's axiom implies that all such sets are Ramsey [17]. Furthermore, large-cardinal axioms, or determinacy axioms, imply that all projective, or doi:10.1006 aima.2001.1983, available online at http: www.idealibrary.com on

Indestructible properties of S- and L-spaces

Building on a method of Abraham and Todorčevič we prove a preservation theorem on certain properties under c.c.c. forcings. Applying this result we show that (1) an uncountable, first countable, 0-dimensional space containing only countable and co-countable open subspaces, and (2) S- and L-groups can exist under Martin's Axiom.

Lusin sequences under CH and under Martin's Axiom

Assuming the continuum hypothesis there is an inseparable sequence of length $\omega _1$ that contains no Lusin subsequence, while if Martin's Axiom and $\neg \rm CH$ are assumed then every inseparable sequence (of length $\omega _1$) is a union of counta

Chain conditions in maximal models

We present two ${\mathbb P}_{\max}$ varations which create maximal models relative to certain counterexamples to Martin's Axiom, in hope of separating certain classical statements which fall between MA and Suslin's Hypothesis. One of these models is taken

James E. Baumgartner. Sacks forcing and the total failure of Martin's axiom. Topology and its applications, vol. 19 (1985), pp. 211–225. - James E. Baumgartner and Peter Dordal. Adjoining dominating functions. The journal of symbolic logic, vol. 50 (1985), pp. 94–101.

James E. Baumgartner. Sacks forcing and the total failure of Martin's axiom. Topology and its applications, vol. 19 (1985), pp. 211–225. - James E. Baumgartner and Peter Dordal. Adjoining dominating functions. The journal of symbolic logic, vol. 50 (1985), pp. 94–101. - Volume 6 Issue 4

Residual measures in locally compact spaces

A σ -finite diffused Borel measure in a topological space is called residual if each nowhere dense set has measure zero. If the measure is also fully supported, then it is called normal . Results on the influence of Martin's Axiom and the Continuum Hypothesis on the existence of residual and normal measures in locally compact spaces are obtained. A connection with L -spaces is established.

Bounded forcing axioms as principles of generic absoluteness

We show that Bounded Forcing Axioms (for instance, Martin's Axiom, the Bounded Proper Forcing Axiom, or the Bounded Martin's Maximum) are equivalent to principles of generic absoluteness, that is, they assert that if a $\Sigma_1$ sentence of the language of set theory with parameters of small transitive size is forceable, then it is true. We also show that Bounded Forcing Axioms imply a strong form of generic absoluteness for projective sentences, namely, if a $\Sigma^1_3$ sentence with parameters is forceable, then it is true. Further, if for every real x, $x^{\sharp}$ exists, and the second uniform indiscernible is less than $\omega_2$ , then the same holds for $\Sigma^1_4$ sentences.

Martin's Axiom and the Dual Distributivity Number

We show that it is consistent that Martin's axiom holds, the continuum is large, and yet the dual distributivity number ℌ is κ1. This answers a question of Halbeisen.

A Space with Only Borel Subsets

Miklos Laczkovich asked if there exists a Haussdorff (or even normal) space in which every subset is Borel yet it is not meager. The motivation of the last condition is that under MA_kappa every subspace of the reals of cardinality kappa has the property that all subsets are F_sigma, however Martin's axiom also implies that these subsets are meager. Here we answer Laczkovich' question.

Set Ideals with Complete Symmetry Group and Partition Ideals

For a wide class of set ideals (including, for example, all uniform ideals), a criterion of completeness of their symmetry groups is provided in terms of ideal quotients (polars). We apply it to partition ideals, and derive the extended Sierpiński–Erdös duality principle. We demonstrate that if the measure and category ideals I0 and I1 on the real line R are partition (or, equivalently, if just I0 is partition), then not only are they isomorphic via an involution, but they also have complete (and distinct) symmetry groups coinciding, respectively, with the symmetry groups of the polars I 0 ⊥ and I 1 ⊥ . The measure and category ideals on R (and in more general spaces) are partition (Oxtoby) ideals assuming Martin's Axiom. In this case their polars are, respectively, the ideals generated by c-Sierpiński and c-Lusin sets. It is well known that the isomorphism of the measure and category ideals is not provable in ZFC. We show that the isomorphism of their symmetry groups is likewise unprovable. 1991 Mathematics Subject Classification 20B35, 03E50, 04A20.

Dimension zero vs measure zero

The following problem is discussed: If X is a topological space of universal measure zero, does it have also dimension zero? It is shown that in a model of set theory it is so for separable metric spaces and that under the Martin's Axiom there are separable metric spaces of positive dimension yet of universal measure zero. It is also shown that for each finite measure in a metric space there is a zero-dimensional subspace that has full measure. Similar questions concerning perfectly meager sets and other types of small sets are also discussed.

Topological completeness of spaces of measures

We prove that the functors and of Radon and -additive probability measures, respectively, preserve neither the real-completeness nor the Dieudonne completeness of Tychonoff spaces. We suggest conditions under which Martin's axiom implies that preserves real-complete spaces, absolute extensors, and Tychonoff bundles. These last results cannot be obtained without additional set-theoretic assumptions.

On a preservation of completeness of uniform spaces by the functor PscT

It is shown that the functor P scT of t -additive probability measures does not preserve completeness of uniform spaces. Namely, P T( R c) is not complete. On the other hand, assuming Martin's Axiom, P scT ( X) is complete for every complete uniform space X of uniform weight < c.

Topological groups in which each nowhere dense subset is closed

Assuming the validity of the combinatorial principlep=c, which follows from Martin's axiom, it is proved that an arbitrary nondiscrete metrizable group topology on an Abelian group can be strengthened to a nondiscrete group topology in which each nowhere dense subset is closed.

Suitable sets for topological groups

A subset S of a topological group G is said to be a suitable set if (a) it has the discrete topology, (b) it is a closed subset of G{1}, and (c) the subgroup generated by S is dense in G. K.H. Hoffmann and S.A. Morris proved that every locally compact group has a suitable set. In this paper it is proved that every metrizable topological group and every countable Hausdorff topological group has a suitable set. Examples of Hausdorff topological groups without suitable sets are produced. The free abelian topological group on the Stone-Čech compactification of any uncountable discrete space is one such example. Under the assumption of the Continuum Hypothesis or Martin's Axiom it is shown that examples exist of separable Hausdorff topological groups with no suitable set. It is not known if such examples exist in ZFC alone. An example is produced here of a compact connected abelian group with a one-element suitable set which has a dense σ-compact connected subgroup with no suitable set.

Completely Additive Liftings

The purpose of this communication is to survey a theory of liftings, as developed in author's thesis ([8]). The first result in this area was Shelah's construction of a model of set theory in which every automorphism ofP(ℕ)/ Fin, where Fin is the ideal of finite sets, istrivial, or inother words, it is induced by a function mapping integers into integers ([33]). (It is a classical result of W. Rudin [31] that under the Continuum Hypothesis there are automorphisms other than trivial ones.) Soon afterwards, Velickovic ([47]), was able to extract from Shelah's argument the fact that every automorphism ofP(ℕ)/ Fin with a Baire-measurable lifting has to be trivial. This, for instance, implies that in Solovay's model ([36]) all automorphisms are trivial. Later on, an axiomatic approach was adopted and Shelah's conclusion was drawn first from the Proper Forcing Axiom (PFA) ([34]) and then from the milder Open Coloring Axiom (OCA) and Martin's Axiom (MA) ([48], see §5 for definitions). Both shifts from the quotientP(ℕ)/ Fin to quotients over more general idealsP(ℕ)/Iand from automorphisms to arbitrary ho-momorphisms were made by Just in a series of papers ([14]-[17]), motivated by some problems in algebra ([7, pp. 38–39], [43, I.12.11], [45, Q48]) and topology ([46, p. 537]).

Sticks and clubs

We study combinatorial principles known as stick and club. Several variants of these principles and cardinal invariants connected to them are also considered. We introduce a new kind of side by-side product of partial orderings which we call pseudo-product. Using such products, we give several generic extensions where some of these principles hold together with ¬CH and Martin's axiom for countable p.o.-sets. An iterative version of the pseudo-product is used under an inaccessible cardinal to show the consistency of the club principle for every stationary subset of limits of ω 1 together with ¬CH and Martin's axiom for countable p.o.-sets.

Simple forcing notions and forcing axioms

In the present paper we are interested in simple forcing notions and Forcing Axioms. A starting point for our investigations was the article [4] in which several problems were posed. We answer some of those problems here.In the first section we deal with the problem of adding Cohen reals by simple forcing notions. Here we interpret simple as of small size. We try to establish as weak as possible versions of Martin Axiom sufficient to conclude that some forcing notions of size less than the continuum add a Cohen real. For example we show that MA(σ-centered) is enough to cause that every small σ-linked forcing notion adds a Cohen real (see Theorem 1.2) and MA(Cohen) implies that every small forcing notion adding an unbounded real adds a Cohen real (see Theorem 1.6). A new almost ωω-bounding σ-centered forcing notion ℚ⊚ appears naturally here. This forcing notion is responsible for adding unbounded reals in this sense, that MA(ℚ⊚) implies that every small forcing notion adding a new real adds an unbounded real (see Theorem 1.13).In the second section we are interested in Anti-Martin Axioms for simple forcing notions. Here we interpret simple as nicely definable. Our aim is to show the consistency of AMA for as large as possible class of ccc forcing notions with large continuum. It has been known that AMA(ccc) implies CH, but it has been (rightly) expected that restrictions to regular (simple) forcing notions might help.

Extensions of functions in Mrówka-Isbell spaces

For an almost disjoint family (a.d.f.) ∑ of subsets of ω, let Ψ(∑) be the Mrówka-Isbell space on ∑. In this article we will analyze the following problem: given an a.d.f. ∑ and a function φ: ∑ → {0, 1} (respectively φ: ∑ → R ) is it possible to extend φ continuously to a big enough subspace ∑ ∪ N of Ψ(∑) for which l Ψ(∑) N ⊃ ∑? Such an extension is called essential. We will prove that: 1. (i) for every a.d.f. ∑ of cardinality 2 ℵ 0 we can find a function φ: ∑ → {0, 1} without essential extensions; 2. (ii) for every m.a.d. family ∑ there exists a function φ: ∑ → R that has no essential extension; and 3. (iii) there exists a Mrówka-Isbell space Ψ(∑) of cardinality ℵ 1 such that every function φ: ϵ → R with at least two different uncountable fibers, has no full extension. On the other hand, under Martin's Axiom every function φ: ∑ → {0, 1} (respectively φ: ∑ → R ) has an essential extension if ¦∑¦ < 2 ℵ 0 . Finally, we analyze these questions under CH and by adding new Cohen reals to a ground model M showing that the existence of an uncountable a.d.f. ∑ for which every onto function φ: ∑ → {0, 1} with infinite fibers has no essential extensions is consistent with ZFC.