On a question of Maarten Maurice

In this note we give ZFC results that reduce the question of Maarten Maurice about the existence of σ-closed-discrete dense subsets of perfect generalized ordered spaces to the study of very special Baire spaces, and we discuss the current status of the question for spaces with small density. Work of Shelah, Todor?evic, Qiao, and Tall shows that Maurice's problem is undecidable for generalized ordered spaces of local density ω₁.     

The consistencies ofMA,SOCA, OCA andISA withKT(ω2)

In this paper, we prove that ifZFC is consistent, then so are the following theories: $$\begin{gathered} ZFC + MA + KT(\omega _2 ) + 2^{\aleph _0 } = \aleph _2 , \hfill \\ ZFC + SOCA + KT(\omega _2 ), \hfill \\ ZFC + SOCA1 + KT(\omega _2 ), \hfill \\ ZFC + OCA + KT(\omega _2 ), \hfill \\ ZFC + ISA + KT(\omega _2 ), \hfill \\ \end{gathered} $$ whereMA denotes Martin's axiom.KT(ω ₂) the statement:“There exists anω ₂-Kurepa tree”, andSOCA, SOCA1,OCA andISA are axioms introduced in [1].     

On topological spaces of singular density and minimal weight

We introduce a weakening of the generalized continuum hypothesis, which we will refer to as the prevalent singular cardinals hypothesis, and show it implies that every topological space of density and weight ℵω₁ is not hereditarily Lindelöf.The assumption PSH is very weak, and in fact holds in all currently known models of ZFC.     

Lindelöf models of the reals: Solution to a problem of Sikorski

We show that it is consistent with ZFC that there is a modelM of ZF + DC such that the integers ofM areω ₁-like, the reals ofM have cardinalityω ₂, and the unit interval [0, 1] ^M is Lindelöf (i.e. every open cover has a countable subcover). This answers an old question of Sikorski.     

The Consistency Strength of \aleph_{\omega} and \aleph_{{\omega}_1} Being Rowbottom Cardinals Without the Axiom of Choice

We show that for all natural numbers n, the theory “ZF + DC $_{\aleph_n}$ + $\aleph_{\omega}$ is a Rowbottom cardinal carrying a Rowbottom filter” has the same consistency strength as the theory “ZFC + There exists a measurable cardinal”. In addition, we show that the theory “ZF + $\aleph_{\omega_1}$ is an ω ₂-Rowbottom cardinal carrying an ω ₂-Rowbottom filter and ω ₁ is regular” has the same consistency strength as the theory “ZFC + There exist ω ₁ measurable cardinals”. We also discuss some generalizations of these results.     

Forcing axioms and the continuum hypothesis

Woodin has demonstrated that, in the presence of large cardinals, there is a single model of ZFC which is maximal for Π₂-sentences over the structure (H(ω ₂), ∈, NS _ω1), in the sense that its (H(ω ₂), ∈, NS _ω1) satisfies every Π₂-sentence σ for which (H(ω ₂), ∈, NS _ω1) ? σ can be forced by set-forcing. In this paper we answer a question of Woodin by showing that there are two Π₂-sentences over the structure (H(ω ₂), ∈, ω ₁) which can each be forced to hold along with the continuum hypothesis, but whose conjunction implies $ {2^{{{\aleph_0}}}}={2^{{{\aleph_1}}}} $ . In the process we establish that there are two preservation theorems for not introducing new real numbers by a countable support iterated forcing which cannot be subsumed into a single preservation theorem.     

Wide scattered spaces and morasses

We show that it is relatively consistent with ZFC that ω² is arbitrarily large and every sequence s=〈sα:α<ω₂〉 of infinite cardinals with sα?ω² is the cardinal sequence of some locally compact scattered space.     

Resolvability vs. almost resolvability

A space X is κ-resolvable (resp. almost κ-resolvable) if it contains κ dense sets that are pairwise disjoint (resp. almost disjoint over the ideal of nowhere dense subsets of X).Answering a problem raised by Juhász, Soukup, and Szentmiklóssy, and improving a consistency result of Comfort and Hu, we prove, in ZFC, that for every infinite cardinal κ there is an almost κ²-resolvable but not ω₁-resolvable space of dispersion character κ.     

More results in polychromatic Ramsey theory

We study polychromatic Ramsey theory with a focus on colourings of [ω ₂]². We show that in the absence of GCH there is a wide range of possibilities. In particular each of the following is consistent relative to the consistency of ZFC: (1) 2 ^ω = ω ₂ and \(\omega _2 \to ^{poly} (\alpha )_{\aleph _0 - bdd}^2 \) for every α <ω ₂; (2) 2 ^ω = ω ₂ and \(\omega _2 \nrightarrow ^{poly} (\omega _1 )_{2 - bdd}^2 \).     

Arcs in the plane

Assuming PFA, every uncountable subset E of the plane meets some C¹ arc in an uncountable set. This is not provable from MA(ℵ₁), although in the case that E is analytic, this is a ZFC result. The result is false in ZFC for C² arcs, and the counter-example is a perfect set.     

Trivial automorphisms

We prove that the statement ‘For all Borel ideals I and J on ω, every isomorphism between Boolean algebras P(ω)/I and P(ω)/J has a continuous representation’ is relatively consistent with ZFC. In this model every isomorphism between P(ω)/I and any other quotient P(ω)/J over a Borel ideal is trivial for a number of Borel ideals I on ω. We can also assure that the dominating number, σ, is equal to ?₁ and that \({2^{{\aleph _1}}} > {2^{{\aleph _0}}}\) . Therefore, the Calkin algebra has outer automorphisms while all automorphisms of P(ω)/Fin are trivial. Proofs rely on delicate analysis of names for reals in a countable support iteration of Suslin proper forcings.     

On the product of homogeneous spaces

Within the class of Tychonoff spaces, and within the class of topological groups, most of the natural questions concerning ‘productive closure’ of the subclasses of countably compact and pseudocompact spaces are answered by the following three well-known results: (1) [ZFC] There is a countably compact Tychonoff space X such that X × X is not pseudocompact; (2) [ZFC] The product of any set of pseudocompact topological groups is pseudocompact; and (3) [ZFC+ MA] There are countably compact topological groups G₀, G₁ such that G₀ × G₁ is not countably compact.In this paper we consider the question of ‘productive closure” in the intermediate class of homogeneous spaces. Our principal result, whose proof leans heavily on a simple, elegant result of V.V. Uspenski?, is this: In ZFC there are pseudocompact, homogeneous spaces X₀, X₁ such that X₀ × X₁ is not pseudocompact; if in addition MA is assumed, the spaces X_i may be chosen countably compact.Our construction yields an unexpected corollary in a different direction: Every compact space embeds as a retract in a countably compact, homogeneous space. Thus for every cardinal number α there is a countably compact, homogeneous space whose Souslin number exceeds α.     

Characterizingω 1 and the long line by their topological elementary reflections

Given a topological space 〈X, T〉 ∈M, an elementary submodel of set theory, we defineX _Mto beX ∩M with the topology generated by {U ∩M : U ∈T ∩M}. We prove that it is undecidable whetherX _Mhomeomorphic toω ₁ impliesX =X _M,yet it is true in ZFC that ifX _Mis homeomorphic to the long line, thenX =X _M.The former result generalizes to other cardinals of uncountable confinality while the latter generalizes to connected, locally compact, locally hereditarily LindelöfT ₂ spaces.     

Examples from trees, related to discrete subsets, pseudo-radiality and ω-boundedness

We construct some examples using trees. Some of them are consistent counterexamples for the discrete reflection of certain topological properties. All the properties dealt with here were already known to be non-discretely reflexive if we assume CH and we show that the same is true assuming the existence of a Suslin tree. In some cases we actually get some ZFC results. We construct also, using a Suslin tree, a compact space that is pseudo-radial but it is not discretely generated. With a similar construction, but using an Aronszajn tree, we present a ZFC space that is first countable, ω-bounded but is not strongly ω-bounded, answering a question of Peter Nyikos.     

A two-coloring of Cartesian products

We shall color the Cartesian product ω × ω₁with two colors. Can an infinite subset A ?ω and an uncountable subset B ?ω₁ be found such that the product A × B can be one-colored? This problem proves to be unsolvable in ZFC.     

Fodor-type Reflection Principle and reflection of metrizability and meta-Lindelöfness

We introduce a new reflection principle which we call “Fodor-type Reflection Principle” (FRP). This principle follows from but is strictly weaker than Fleissner's Axiom R. For instance, FRP does not impose any restriction on the size of the continuum, while Axiom R implies that the continuum has size ?ℵ₂.We show that FRP implies that every locally separable countably tight topological space X is meta-Lindelöf if all of its subspaces of cardinality ?ℵ₁ are (Theorem 4.3). It follows that, under FRP, every locally (countably) compact space is metrizable if all of its subspaces of cardinality ?ℵ₁ are (Corollary 4.4). This improves a result of Balogh who proved the same assertion under Axiom R.We also give several other results in this vein, some in ZFC, others in some further extension of ZFC. For example, we prove in ZFC that if X is a locally (countably) compact space of singular cardinality in which every subspace of smaller size is metrizable then X itself is also metrizable (Corollary 5.2).     

Hyperspace of convex compacta of nonmetrizable compact convex subspaces of locally convex spaces

Our main result states that the hyperspace of convex compact subsets of a compact convex subset X in a locally convex space is an absolute retract if and only if X is an absolute retract of weight ?ω₁. It is also proved that the hyperspace of convex compact subsets of the Tychonov cube Iω₁ is homeomorphic to Iω₁. An analogous result is also proved for the cone over Iω₁. Our proofs are based on analysis of maps of hyperspaces of compact convex subsets, in particular, selection theorems for such maps are proved.     

Localizing the axioms

We examine what happens if we replace ZFC with a localistic/relativistic system, LZFC, whose central new axiom, denoted by Loc(ZFC), says that every set belongs to a transitive model of ZFC. LZFC consists of Loc(ZFC) plus some elementary axioms forming Basic Set Theory (BST). Some theoretical reasons for this shift of view are given. All ${\Pi_2}We examine what happens if we replace ZFC with a localistic/relativistic system, LZFC, whose central new axiom, denoted by Loc(ZFC), says that every set belongs to a transitive model of ZFC. LZFC consists of Loc(ZFC) plus some elementary axioms forming Basic Set Theory (BST). Some theoretical reasons for this shift of view are given. All P₂{\Pi_2} consequences of ZFC are provable in LZFC. LZFC strongly extends Kripke-Platek (KP) set theory minus Δ₀-Collection and minus ?{\in} -induction scheme. ZFC+ “there is an inaccessible cardinal” proves the consistency of LZFC. In LZFC we focus on models rather than cardinals, a transitive model being considered as the analogue of an inaccessible cardinal. Pushing this analogy further we define α-Mahlo models and P₁¹{\Pi_1^1} -indescribable models, the latter being the analogues of weakly compact cardinals. Also localization axioms of the form Loc(ZFC+f){Loc({\rm ZFC}+\phi)} are considered and their global consequences are examined. Finally we introduce the concept of standard compact cardinal (in ZFC) and some standard compactness results are proved.     

The stationary set splitting game

The stationary set splitting game is a game of perfect information of length ω₁ between two players, unsplit and split, in which unsplit chooses stationarily many countable ordinals and split tries to continuously divide them into two stationary pieces. We show that it is possible in ZFC to force a winning strategy for either player, or for neither. This gives a new counterexample to Σ²₂ maximality with a predicate for the nonstationary ideal on ω₁, and an example of a consistently undetermined game of length ω₁ with payoff de.nable in the second‐order monadic logic of order. We also show that the determinacy of the game is consistent with Martin's Axiom but not Martin's Maximum. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

o-Boundedness of free topological groups

Assuming the absence of Q-points (which is consistent with ZFC) we prove that the free topological group F(X) over a Tychonov space X is o-bounded if and only if every continuous metrizable image T of X satisfies the selection principle fin_?(O,Ω) (the latter means that for every sequence 〈un〉_n∈ω of open covers of T there exists a sequence 〈vn〉_n∈ω such that vn∈[un]^<ω and for every F∈[X]^<ω there exists n∈ω with F⊂?vn). This characterization gives a consistent answer to a problem posed by C. Hernándes, D. Robbie, and M. Tkachenko in 2000.