Martin's Axiom Research Articles

Sacks forcing, Laver forcing, and Martin's axiom

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.

Internal forcing axioms: Martin's axiom and the proper forcing axiom

Internal forcing axioms: Martin's axiom and the proper forcing axiom

Separation in Ψ-spaces

A Ψ-space is the topological space usually associated with a maximal almost disjoint family of subsets of the integers. In this paper, we study properties that an infinite set of nonisolated points in a Ψ-space might possess. These properties are 2-embeddedness, C ∗-embeddedness, and another weaker property, that of being solidly normalized. It is known from [2,3] that the existence of a Ψ-space with an infinite set of nonisolated points with any of these properties is independent of ZFC. Here we show that these three properties are distinct by providing examples, assuming less than Martin's axiom, of Ψ-spaces with infinite sets of nonisolated points which in one case are C ∗- but not 2-embedded, and in the other case solidly normalized but not C ∗-embedded. Additionally, Martin's axiom implies the existence of a Ψ-space with a set S of nonisolated points of cardinality c such that every subset of S with cardinality less than c is 2-embedded.

Remarks on Martin's Axiom and the Continuum Hypothesis

Martin's axiom and the Continuum Hypothesis are studied here using the notion of accc partitioni.e., a partition of the formwhereK0has the following properties:(a)K0contains subsets of its elements as well as all singletons ofX.(b) Every uncountable subset of K0contains two elements whose union is inK0.

A strong Baire category theorem for products

The notion of a set's depending on a given coordinate in a product space is briefly developed, and the following generalization of the Baire Category Theorem is proved: when D is a family of dense open subsets of a product of pseudocomplete spaces such that each coordinate has countably many sets of D depending on it, then D has dense intersection. Under Martin's Axiom, the hypothesis “countably many” may be replaced by “less than continuum many”.

Nonnormality or hereditary paracompactness of some spaces of real functions

The topology we give a set F of real functions is determined not by ε-balls, with ε>0, but by e-balls, with e ϵ F strictly positive. For F we will consider C( X), the continous real functions on a space X, or L and Ba, the real functions on R which are Lebesgue, or Baire measurable; in this case we are interested in the quotient spaces L/∼, and Ba/∼, with f∼ g if f agrees with g except on a null set, or on a meager set. We first show that C R is not normal, and then generalize this and prove nonnormality for nondiscrete, first countable or locally compact, nonpseudocompact X. We find a nonpseudocompact space S without isolated points such that C( S) is hereditarily paracompact under CH, the Continuum Hypothesis. We next show that L/∼ is hereditarily paracompact under MA, Martin's Axiom, but that Ba/∽ is not hereditarily normal. Also, L/∼ and Ba/∼ are proved nonhomeomorphic without using MA. We also consider a topology on C( X) induced by open subsets of X× R .

Representation 0f certain functions under martin's axiom

Representation 0f certain functions under martin's axiom

Extending discrete-valued functions

In this paper, we show that for a separable metric space X, every continuous function from a subset S of X into a finite discrete space extends to a continuous function on X if and only if every continuous function from S into any discrete space extends to a continuous function on X. We also show that if there is no inner model having a measurable cardinal, then there is a metric space X with a subspace S such that every 2-valued continuous function from S extends to a continuous function on all of X, but not every discrete-valued continuous function on S extends to such a map on X. Furthermore, if Martin's Axiom is assumed, such a space can be constructed so that not even co-valued functions on S need extend. This last result uses a version of the Isbell-Mrowka space 'P having a C· -embedded infinite discrete subset. On the other hand, assuming the Product Measure Extension Axiom, no such 'P exists.

The Product of a Lindelof Space with the Space of Irrationals Under Martin's Axiom

We give an example of a Lindelof space whose product with the space of irrationals is not Lindelof provided that Martin's axiom holds. Our results is an improvement of a Michael's construction using the Continuum Hypothesis

Set theoretic properties of Loeb measure

AbstractIn this paper we ask the question: to what extent do basic set theoretic properties of Loeb measure depend on the nonstandard universe and on properties of the model of set theory in which it lies? We show that, assuming Martin's axiom and κ-saturation, the smallest cover by Loeb measure zero sets must have cardinality less than κ. In contrast to this we show that the additivity of Loeb measure cannot be greater than ω1. Define cof(H) as the smallest cardinality of a family of Loeb measure zero sets which cover every other Loeb measure zero set. We show that card(⌊log2(H)⌋) ≤ cof (H) ≤ card(2H), where card is the external cardinality. We answer a question of Paris and Mills concerning cuts in nonstandard models of number theory. We also present a pair of nonstandard universes M ≼ N and hyperfinite integer H ∈ M such that H is not enlarged by N, 2H contains new elements, but every new subset of H has Loeb measure zero. We show that it is consistent that there exists a Sierpiński set in the reals but no Loeb-Sierpiński set in any nonstandard universe. We also show that it is consistent with the failure of the continuum hypothesis that Loeb-Sierpiński sets can exist in some nonstandard universes and even in an ultrapower of a standard universe.

A class of consistent anti-Martin’s axioms

Both the Continuum Hypothesis and Martin's Axiom allow inductive constructions to continue in circumstances where the inductive hypothesis might otherwise fail. The search for useful related axioms procedes naturally in two directions: towards «Super Martin's Axioms», which extend MA to broader classes of orders; and towards «Anti-Martin's Axioms» (AMA's) which are strictly weaker than CH, but which, when combined with −CH, deny MA. We consider restrictions of van Douwen and Fleissner's Undefinable Forcing Axiom which are consistent with the negation of the continuum hypothesis

Provable forms of Martin's axiom.

It is shown that if Martin's Axiom (MA) is restricted to well-orderable sets, and if the countable antichain condition is replaced by either a finite antichain condition or a finite «Q-strong antichain» condition, then the resulting statements are forms of MA provable in ZF. Variations of these weak forms of MA are shown to be equivalent to the Axiom of Choice and some of its weak forms

Definable forcing axiom: an alternative to Martin's axiom

Definable forcing axiom: an alternative to Martin's axiom

Martin's axiom and ordinal graphs: large

From set mapping theorems, Erd ' ́ os, Hajnal and Milner proved that every graph on a limit ordinal α< ω ω+2 1 either contains an independentset of type α or an infinite path. In this paper, Martin's Axiom is used to extend the positive result to all limit ordinals less than the continuum. In another paper, under the assumption of Jensen's Diamond Principle, Baumgartner and Larson have constructed counter-examples for ordinals α with ω ω+1 1< α< ω 2

A diamond example of an ordinal graph with no infinite paths

From set mapping theorems, Erd ' ́ os, Hajnal and Milner proved that every graph on a limit ordinal α < ω ω+2 1 either contains an independent set of type α or an infinite path. In symbols one writes α → (α, infinite path) 2. In this paper the assumption of Jensen's Diamond Principle is used to construct counter-examples for α with ω ω+2 1 ⩽ α < ω 2. Under Martin's Axiom, Larson has shown no such counter-examples exist for limit α below the continuum.

Martin's Axiom implies a stronger version of Blumberg's Theorem

Martin's Axiom implies a stronger version of Blumberg's Theorem

Martin's axiom and pathological points in β X⧹ X

P(c), a well-known consequence of Martin's Axiom, implies that if X is a nonpseudocompact space with countable π-weight, then there is a point p in β X⧹ X such that: (1) p is not in the closure of any nowhere dense subset of X (i.e., p is a remote point), (2) p is not simultaneously in the closure of any two disjoint open subsets of β X (i.e., β X is extremally disconnected at p), and (3) if X is locally compact, then each intersection of fewer than c neighborhoods of p in β X⧹ X is again a neighborhood of p (i.e., p is a P c-point). The construction of such points depends on the fact that P(c) is equivalent to the proposition that if X is any noncompact realcompact locally compact space with countable π-weight, then each nonempty intersection of fewer than c open subsets of β X⧹ X has a nonempty interior. The condition on the π-weight is essential: There is a noncompact σ-compact (hence realcompact) locally compact separable space M such that each point of β M⧹ M is contained in an intersection of ω 1 open sets, which has empty interior. M has the remarkable property that β M⧹ M and β N ⧹ N are homeomorphic under CH, the Continuum Hypothesis, but not under P(c) + ¬op;CH. P(c) also implies that if X is a noncompact realcompact locally compact space with countable π-weight, then each point of β X⧹ X is simultaneously in the closure of c pairwise disjoint open subsets of β X⧹ X (i.e., is a c-point of β X⧹ X). Finally, if ƒ: β Q →β R is the (unique) continuous map such that ƒ↾ Q = id Q , then P(c) implies the existence of a subset E of β R ⧹ R with cardinality 2 c, such that ƒ ←[{x}] is a one-point set for each x ϵ E(hence ƒ↾( Q∪ƒ ←[E]) is a homeomorphism).

On the generic existence of special ultrafilters

We introduce the concept of the generic existence of P-point, Qpoint, and selective ultrafilters, a concept which is somewhat stronger than the existence of these sorts of ultrafilters. We show that selective ultrafilters exist generically if semiselectives do iff in -= c , and we show that Q-point ultrafilters exist generically if semiQ-points do iff rnc = d, where d is the minimal cardinality of a dominating family of functions and mc is the minimal cardinality of a cover of the real line by nowhere-dense sets. These results complement a result of Ketonen, that P-points exist generically iff c d, and one of P. Nyikos and D. H. Fremlin, that saturated ultrafilters exist generically if c= 2 g(n) . Following [21] we let d denote the minimum cardinality of a dominating family of functions. We let c be the cardinality 'aw and mc be the minimum cardinality of a cover of the real line consisting of nowhere-dense sets. The notation mc comes from Martin's Axiom for countable partial orders; mc is the maximum cardinal A such that the following holds: given any countable partial order P and fewer than 2 dense subsets of P, there exists a generic G C P which meets each of the dense subsets. Also, mn may be characterized as the maximum cardinal 2 such that, given any family of functions H of size < i, we may find a function g such that Vh c H, 3n E co g(n) = h(n). Some of the basic properties of this cardinal, including proofs of the equivalence of these characterizations, may be found in [1, [1 5], and [22]. It is clear from the above discussion that Mc : d < c. In this paper we will consider the question of the existence of various special sorts of ultrafilters, whose definitions we now mention. An ultrafilter U is called a P-point if every function is either finite-to-one or bounded on a set in U. An ultrafilter U is called a Q-point or rare ultrafilter if every finite-to-one function is one-to-one on a set in U. An ultrafilter U is said to be a semiQ-point (or Received by the editors July 2, 1988 and, in revised form, May 1, 1989. 1980 Mathemnatics Subject Classification (1985 Revision). Primary 03E05; Secondarv 03E35, 54H05. ? 1990 Akmerican Mathematical Society 0002-9939/90 $1.00 + $.25 per page

Saturating ultrafilters on N

AbstractWe discuss saturating ultrafilters on N, relating them to other types of non-principal ultrafilter. (a) There is an (ω,c)-saturating ultrafllter on N iff 2λ≤ c for everyλ &lt;c and there is no cover ofRby fewer than c nowhere dense sets, (b) Assume Martin's axiom. Then, for any cardinalκ, a nonprincipal ultrafllter on N is (ω, κ)-saturating iff it is almostκ-good. In particular, (i)p(κ)-point ultrafilters are (ω, κ)-saturating, and (ii) the set of (ω, κ)-saturating ultrafilters is invariant under homeomorphisms ofβN/N. (c) It is relatively consistent with ZFC to suppose that there is a Ramseyp(c)-point ultrafilter which is not (ω, c)-saturating.

Is the product of CCC spaces a CCC space?

In this expository paper it is shown that Martin's Axiom and the negation of the Continuum Hypothesis imply that the product of ccc spaces is a ccc space. The Continuum Hypothesis is then used to construct the Laver-Gavin example of two ccc spaces whose product is not a ccc space.