Martin's Axiom Research Articles

MA and three Fréchet spaces

M. Scheepers introduced the notion of selectively separable as one of many interesting selection principles. Fréchet spaces are known to be selectively separable but neither of these properties is well-behaved in products, even for countable spaces. We prove that the open coloring axiom, OCA, implies that the product of two countable Fréchet spaces is selectively separable. We prove that Martin's Axiom implies there are three countable Fréchet spaces whose product is not selectively separable.

Wetzel families and the continuum

AbstractWe provide answers to a question brought up by Erdős about the construction of Wetzel families in the absence of the continuum hypothesis: A Wetzel family is a family of entire functions on the complex plane which pointwise assumes fewer than values. To be more precise, we show that the existence of a Wetzel family is consistent with all possible values of the continuum and, if is regular, also with Martin's Axiom. In the particular case of this answers the main open question asked by Kumar and Shelah [Fund. Math. 239 (2017) no. 3, 279–288]. In the buildup to this result, we are also solving an open question of Zapletal on strongly almost disjoint functions from Zapletal [Israel J. Math. 97 (1997) no. 1, 101–111]. We also study a strongly related notion of sets exhibiting a universality property via mappings by entire functions and show that these consistently exist while the continuum equals .

Two chain conditions and their Todorčević's fragments of Martin's Axiom

In this paper, we investigate two chain conditions of forcing notions, called the rectangle refining property and property R1,ℵ1. They are stronger than the countable chain condition. Some of their typical examples are forcing notions about indestructible gaps introduced by Kunen. Both chain conditions are similar and have common examples, however, no distinction between them is known so far.In this paper, we propose a difference of two chain conditions from the view point of Todorčević's fragments of Martin's Axiom. More precisely, it is proved that some Todorčević's fragment of Martin's Axiom for forcing notions with property R1,ℵ1 is consistent with the existence of a non-special Aronszajn tree, and that some Todorčević's fragment of Martin's Axiom for forcing notions with the rectangle refining property in some weak sense implies that every Aronszajn tree is special.

The relative strengths of fragments of Martin's axiom

We give a survey of results on the relative strengths of different fragments of Martin's Axiom, as well as a list of the main remaining open questions.

Differentially compact spaces

We generalize some known results regarding topological spaces defined by combinatorial ideals on ω.(1)We show that no Mrówka space is a differentially compact space.(2)Under the continuum hypothesis, we construct an I-space (where I is a P+-ideal) which is not a differentially compact space.(3)Under Martin's axiom, we construct an I-space (where I is an Fσ ideal) which is not a differentially compact space.

Common hypercyclic vectors for unilateral weighted shifts on ℓ2

Each w∈ℓ∞ defines a bacwards weighted shift Bw:ℓ2→ℓ2. A vector x∈ℓ2 is \textit{hypercyclic} for Bw if the set of forward iterates of x is dense in ℓ2. For each such w, the set HC(w) consisting of all vectors hypercyclic for Bw is Gδ. The set of \textit{common hypercyclic vectors} for a set W⊆ℓ∞ is the set HC∗(W)=⋂w∈WHC(w). We show that HC∗(W) can be made arbitrarily complicated by making W sufficiently complex, and that even for a Gδ set W the set HC∗(W) can be non-Borel. Finally, by assuming the continuum hypothesis or Martin's axiom, we are able to construct a set W such that HC∗(W) does not have the property of Baire.

On the continuous gradability of the cut-point orders of [formula omitted]-trees

An R-tree is a certain kind of metric space tree in which every point can be branching. Favre and Jonsson posed the following problem in 2004: can the class of orders underlying R-trees be characterised by the fact that every branch is order-isomorphic to a real interval? In the first part, I answer this question in the negative: there is a ‘branchwise-real tree order’ which is not ‘continuously gradable’. In the second part, I show that a branchwise-real tree order is continuously gradable if and only if every well-stratified subtree is R-gradable. This link with set theory is put to work in the third part answering refinements of the main question, yielding several independence results. For example, when κ⩾c, there is a branchwise-real tree order which is not continuously gradable, and which satisfies a property corresponding to κ-separability. Conversely, under Martin's Axiom at κ such a tree does not exist.

MA(ℵ0) restricted to complete Boolean algebras and choice

AbstractIt is a long standing open problem whether or not the Axiom of Countable Choice implies the fragment of Martin's Axiom either in or in . In this direction, we provide a partial answer by establishing that the Boolean Prime Ideal Theorem in conjunction with the Countable Union Theorem does not imply restricted to complete Boolean algebras in . Furthermore, we prove that the latter (formally) weaker form of and the Δ‐system Lemma are independent of each other in .We also answer open questions from Tachtsis [16] which concern the status of restricted to complete Boolean algebras in certain Fraenkel–Mostowski permutation models of and we strengthen some results from the above paper.

Selective separability and q+ in maximal spaces

Given a hereditarily meager ideal I on a countable set X we use Martin's axiom for countable posets to produce a zero-dimensional maximal topology τI on X such that τI∩I={∅} and, moreover, if I is p+ then τI is selectively separable (SS) and if I is q+, so is τI. In particular, we obtain regular maximal spaces satisfying all boolean combinations of the properties SS and q+.

Rapid interval P-points

In this paper a rapid non-interval P-point and a rapid non-weakly Ramsey interval P-point are constructed assuming Continuum Hypothesis or Martin's Axiom. These constructions, with the help of geometric interpretations for some needed configurations, provide the examples of rapid ultrafilters for an affirmative answer to a question of Blass, Di Nasso, and Forti.

Isomorphic spectrum and isomorphic length of a Banach space

We prove that, given any ordinal $\delta < \omega_2$, there exists a transfinite $\delta$-sequence of separable Banach spaces $(X_\alpha)_{\alpha < \delta}$ such that $X_\alpha$ embeds isomorphically into $X_\beta$ and contains no subspace isomorphic to $X_\beta$ for all $\alpha < \beta < \delta$. All these spaces are subspaces of the Banach space $E_p = \bigl( \bigoplus_{n=1}^\infty \ell_p \bigr)_2$, where $1 \leq p < 2$. Moreover, assuming Martin's axiom, we prove the same for all ordinals $\delta$ of continuum cardinality.

A countably compact topological group with the non-countably pracompact square

Under Martin's Axiom we construct a Boolean countably compact topological group whose square is not countably pracompact.

Sailing over three problems of Koszmider

We discuss three problems of Koszmider on the structure of the spaces of continuous functions on the Stone compact KA generated by an almost disjoint family A of infinite subsets of ω — we present a solution to two problems and develop previous results of Marciszewski and Pol answering the third one. We will show, in particular, that assuming Martin's axiom the space C(KA) is uniquely determined up to isomorphism by the cardinality of A whenever |A|<c, while there are 2c nonisomorphic spaces C(KA) with |A|=c. We also investigate Koszmider's problems in the context of the class of separable Rosenthal compacta and indicate the meaning of our results in the language of twisted sums of c0 and some C(K) spaces.

Spaces of small cellularity have nowhere constant continuous images of small weight

We call a continuous map f:X→Ynowhere constant if it is not constant on any non-empty open subset of its domain X. Clearly, this is equivalent with the assumption that every fiber f−1(y) of f is nowhere dense in X. We call the continuous map f:X→Ypseudo-open if for each nowhere dense Z⊂Y its inverse image f−1(Z) is nowhere dense in X. Clearly, if Y is crowded, i.e. has no isolated points, then f is nowhere constant.The aim of this paper is to study the following, admittedly imprecise, question: How “small” nowhere constant, resp. pseudo-open continuous images can “large” spaces have? Our main results yield the following two precise answers to this question, explaining also our title. Both of them involve the cardinal function cˆ(X), the “hat version” of cellularity, which is defined as the smallest cardinal κ such that there is no κ-sized disjoint family of open sets in X. Thus, for instance, cˆ(X)=ω1 means that X is CCC.THEOREM A. Any crowded Tychonov space X has a crowded Tychonov nowhere constant continuous image Y of weight w(Y)≤cˆ(X). Moreover, in this statement ≤ may be replaced with < iff there are no cˆ(X)-Suslin lines (or trees).THEOREM B. Any crowded Tychonov space X has a crowded Tychonov pseudo-open continuous image Y of weight w(Y)≤2<cˆ(X). If Martin's axiom holds then there is a CCC crowded Tychonov space X such that for any crowded Hausdorff pseudo-open continuous image Y of X we have w(Y)≥c(=2<ω1).

Thin ultrafilters and the P-hierarchy of ultrafilters

Under Martin Axiom, we prove that for each ordinal γ<ω1 there exists a thin ultrafilter that belongs to the class Pγ of the P-hierarchy of ultrafilters.Since the class P2 of ultrafilters coincides with the class of P-points, this result generalizes a theorem of Flašková, which states that, under the Martin Axiom for countable posets, there exists a thin ultrafilter which is not a P-point. It is also related to a theorem which states that, under Continuum Hypothesis, for any tall P-ideal I on ω there are I-ultrafilters in each class Pγ of the P-hierarchy. However, the ideal of thin sets is not a P-ideal.

Some infinitely generated non-projective modules over path algebras and their extensions under Martin's axiom

In this paper it is proved that, when $Q$ is a quiver that admits some closure, for any algebraically closed field $K$ and any finite dimensional $K$-linear representation $\mathcal{X}$ of $Q$, if ${\rm Ext}^1_{KQ}(\mathcal{X}, KQ) = 0$ then $\mathcal{X}$ is projective. In contrast, we show that if $Q$ is a specific quiver of the type above, then there is an infinitely generated non-projective $KQ$-module $M_{\omega_1}$ such that, when $K$ is a countable field, $\mathbf{MA}_{\aleph_1}$ (Martin's axiom for $\aleph_1$ many dense sets, which is a combinatorial axiom in set theory) implies that ${\rm Ext}^1_{KQ}(M_{\omega_1}, KQ) = 0$.

Parametrized Measuring and Club Guessing

We introduce Strong Measuring, a maximal strengthening of J. T. Moore's Measuring principle, which asserts that every collection of fewer than continuum many closed bounded subsets of $\omega_1$ is measured by some club subset of $\omega_1$. The consistency of Strong Measuring with the negation of CH is shown, solving an open problem from about parametrized measuring principles. Specifically, we prove that Strong Measuring follows from MRP together with Martin's Axiom for $\sigma$-centered forcings, as well as from BPFA. We also consider strong versions of Measuring in the absence of the Axiom of Choice.

𝜎-ideals and outer measures on the real line

Abstract A weak selection on ℝ {\mathbb{R}} is a function f : [ ℝ ] 2 → ℝ {f\colon[\mathbb{R}]^{2}\to\mathbb{R}} such that f ⁢ ( { x , y } ) ∈ { x , y } {f(\{x,y\})\in\{x,y\}} for each { x , y } ∈ [ ℝ ] 2 {\{x,y\}\in[\mathbb{R}]^{2}} . In this article, we continue with the study (which was initiated in [1]) of the outer measures λ f {\lambda_{f}} on the real line ℝ {\mathbb{R}} defined by weak selections f. One of the main results is to show that CH is equivalent to the existence of a weak selection f for which λ f ⁢ ( A ) = 0 {\lambda_{f}(A)=0} whenever | A | ≤ ω {\lvert A\rvert\leq\omega} and λ f ⁢ ( A ) = ∞ {\lambda_{f}(A)=\infty} otherwise. Some conditions are given for a σ-ideal of ℝ {\mathbb{R}} in order to be exactly the family 𝒩 f {\mathcal{N}_{f}} of λ f {\lambda_{f}} -null subsets for some weak selection f. It is shown that there are 2 𝔠 {2^{\mathfrak{c}}} pairwise distinct ideals on ℝ {\mathbb{R}} of the form 𝒩 f {\mathcal{N}_{f}} , where f is a weak selection. Also, we prove that the Martin axiom implies the existence of a weak selection f such that 𝒩 f {\mathcal{N}_{f}} is exactly the σ-ideal of meager subsets of ℝ {\mathbb{R}} . Finally, we shall study pairs of weak selections which are “almost equal” but they have different families of λ f {\lambda_{f}} -measurable sets.

Nontrivial twisted sums for finite height spacesunder Martin’s Axiom

We show that if we assume Martin's Axiom, then there exists a nontrivial twisted sum of c_0 and C(K), for every compact space K with finite height and weight at least continuum. This result settles the problem of existence of nontrivial twisted sums of c_0 and C(K), for finite height spaces K, under Martin's Axiom.

Local extension property for finite height spaces

We introduce a new technique for the study of the local extension property (LEP) for boolean algebras and we use it to show that the clopen algebra of every compact Hausdorff space $K$ of finite height has LEP. This implies, under appropriate additional assumptions on $K$ and Martin's Axiom, that every twisted sum of $c_0$ and $C(K)$ is trivial, generalizing a recent result by Marciszewski and Plebanek.