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.     

Tychonoff expansions with prescribed resolvability properties

The recent literature offers examples, specific and hand-crafted, of Tychonoff spaces (in ZFC) which respond negatively to these questions, due respectively to Ceder and Pearson (1967) [3] and to Comfort and García-Ferreira (2001) [5]: (1) Is every ω-resolvable space maximally resolvable? (2) Is every maximally resolvable space extraresolvable? Now using the method of KID expansion, the authors show that every suitably restricted Tychonoff topological space (X,T) admits a larger Tychonoff topology (that is, an “expansion”) witnessing such failure. Specifically the authors show in ZFC that if (X,T) is a maximally resolvable Tychonoff space with S(X,T)?Δ(X,T)=κ, then (X,T) has Tychonoff expansions U=Ui (1?i?5), with Δ(X,Ui)=Δ(X,T) and S(X,Ui)?Δ(X,Ui), such that (X,Ui) is: (i=1) ω-resolvable but not maximally resolvable; (i=2) [if κ^′ is regular, with S(X,T)?κ^′?κ] τ-resolvable for all τ<κ^′, but not κ^′-resolvable; (i=3) maximally resolvable, but not extraresolvable; (i=4) extraresolvable, but not maximally resolvable; (i=5) maximally resolvable and extraresolvable, but not strongly extraresolvable.     

HFD groups in the Solovay model

Hajnal and Juhász proved that under CH there is a hereditarily separable, hereditarily normal topological group without non-trivial convergent sequences that is countably compact and not Lindelöf. The example constructed is a topological subgroup H⊆ω₁² that is an HFD with the following property

(P)

the projection of H onto every partial product I² for I∈ω[ω₁] is onto.

Any such group has the necessary properties. We prove that if κ is a cardinal of uncountable cofinality, then in the model obtained by forcing over a model of CH with the measure algebra on κ², there is an HFD topological group in ω₁² which has property (P).     

Free Boolean algebras over unions of two well orderings

Given a partially ordered set P there exists the most general Boolean algebra which contains P as a generating set, called the free Boolean algebra over P. We study free Boolean algebras over posets of the form P=P₀∪P₁, where P₀, P₁ are well orderings. We call them nearly ordinal algebras.Answering a question of Maurice Pouzet, we show that for every uncountable cardinal κ there are κ² pairwise non-isomorphic nearly ordinal algebras of cardinality κ.Topologically, free Boolean algebras over posets correspond to compact 0-dimensional distributive lattices. In this context, we classify all closed sublattices of the product (ω₁+1)×(ω₁+1), showing that there are only ℵ₁ many types. In contrast with the last result, we show that there are ℵ₁² topological types of closed subsets of the Tikhonov plank (ω₁+1)×(ω+1).     

Characters of ultrafilters and tightness of products of fans

As applications of productivity of coreflective classes of topological spaces, the following results will be proved: (1) Characters of points of βN?N are not smaller than any submeasurable cardinal less or equal to ω². (2) If κ is a submeasurable cardinal and S is a sequential fan with κ many spines then the tightness of the κ-power of S is equal to κ. In fact, a little more general results are proved.     

New basic result in classical descriptive set theory: Preservation of completeness

The aim of this note is to prove the following result:Assume that f is a continuous function from the space of irrationals ωω onto Y such that the image f(U) of every set U which is open and closed in ωω is the union of one open and one closed set. Then Y is a completely metrizable space.     

Cofinal types of ultrafilters

We study Tukey types of ultrafilters on ω, focusing on the question of when Tukey reducibility is equivalent to Rudin-Keisler reducibility. We give several conditions under which this equivalence holds. We show that there are only c many ultrafilters that are Tukey below any basically generated ultrafilter. The class of basically generated ultrafilters includes all known ultrafilters that are not Tukey above [ω₁]^<ω. We give a complete characterization of all ultrafilters that are Tukey below a selective. A counterexample showing that Tukey reducibility and RK reducibility can diverge within the class of P-points is also given.     

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.     

Rothberger bounded groups and Ramsey theory

We show that:

(1)

Rothberger bounded subgroups of σ-compact groups are characterized by Ramseyan partition relations (Corollary 4).

(2)

For each uncountable cardinal κ there is a T₀ topological group of cardinality κ such that ONE has a winning strategy in the point-open game on the group and the group is not a closed subspace of any σ-compact space (Theorem 8).

(3)

For each uncountable cardinal κ there is a T₀ topological group of cardinality κ such that ONE has a winning strategy in the point-open game on the group and the group is σ-compact (Corollary 17).

     

Cleavability over ordinals

In this paper we show that if X is an infinite compactum cleavable over an ordinal, then X must be homeomorphic to an ordinal. X must also therefore be a LOTS. This answers two fundamental questions in the area of cleavability. We also leave it as an open question whether cleavability of an infinite compactum X over an ordinal λ implies X is embeddable into λ.     

On the Laczkovich-Komjáth property of sigma-ideals

Komjáth in 1984 proved that, for each sequence (An) of analytic subsets of a Polish space X, if lim supn∈HAn is uncountable for every H∈ω[N] then _?n∈GAn is uncountable for some G∈ω[N]. This fact, by our definition, means that the σ-ideal [X]^?ω has property (LK). We prove that every σ-ideal generated by X/E has property (LK), for an equivalence relation E⊂X² of type Fσ with uncountably many equivalence classes. We also show the parametric version of this result. Finally, the invariance of property (LK) with respect to various operations is studied.     

Cascades,order, and ultrafilters

We investigate mutual behavior of cascades, contours of which are contained in a fixed ultrafilter. This allows us to prove (ZFC) that the class of strict _Jωω$J_{{\omega }^{\omega }}$-ultrafilters, introduced by J.E. Baumgartner in [2], is empty. We translate the result to the language of _<∞${<}_{\infty }$-sequences under an ultrafilter, investigated by C. Laflamme in [17], and we show that if there is an arbitrary long finite _<∞${<}_{\infty }$-sequence under u, then u   is at least a strict _Jωω+1$J_{{\omega }^{\omega +1}}$-ultrafilter.     

On a σ-ideal of compact sets

We recall from [T. Mátrai, Kenilworth, Proc. Amer. Math. Soc. 137 (3) (2009) 1115-1125] a Gδσ-ideal of compact subsets of ω² and prove that it is not Tukey reducible to the ideal . This result answers a question of S. Solecki and S. Todor?evi? in the negative.     

Linear orderings and powers of characterizable cardinals

We say that a countable model M completely characterizes an infinite cardinal κ, if the Scott sentence of M has a model in cardinality κ, but no models in cardinality κ⁺. If a structure M completely characterizes κ, κ is called characterizable. In this paper, we concern ourselves with cardinals that are characterizable by linearly ordered structures (cf. Definition 2.1).Under the assumption of GCH, Malitz completely resolved the problem by showing that κ is characterizable if and only if κ=ℵ_α, for some α<ω₁ (cf. Malitz (1968) [7] and Baumgartner (1974) [1]). Our results concern the case where GCH fails.From Hjorth (2002) [3], we can deduce that if κ is characterizable, then κ⁺ is characterizable by a densely ordered structure (see Theorem 2.4 and Corollary 2.5).We show that if κ is homogeneously characterizable (cf. Definition 2.2), then κ is characterizable by a densely ordered structure, while the converse fails (Theorem 2.3).The main theorems are (1) If κ>2^λ is a characterizable cardinal, λ is characterizable by a densely ordered structure and λ is the least cardinal such that κ^λ>κ, then κ^λ is also characterizable (Theorem 5.4) and (2) if ℵ_α and κ^ℵ_α are characterizable cardinals, then the same is true for κ^ℵ_α+β, for all countable β (Theorem 5.5).Combining these two theorems we get that if κ>2^ℵ_α is a characterizable cardinal, ℵ_α is characterizable by a densely ordered structure and ℵ_α is the least cardinal such that κ^ℵ_α>κ, then for all β<α+ω₁, κ^ℵ_β is characterizable (Theorem 5.7). Also if κ is a characterizable cardinal, then κ^ℵ_α is characterizable, for all countable α (Corollary 5.6). This answers a question of the author in Souldatos (submitted for publication) [8].     

On extremally disconnected topological groups

We introduce the notion of a partially selective ultrafilter and prove that (a) if G is an extremally disconnected topological group and p is a converging nonprincipal ultrafilter on G containing a countable discrete subset, then p is partially selective, and (b) the existence of a nonprincipal partially selective ultrafilter on a countable set implies the existence of a P-point in ω^∗. Thus it is consistent with ZFC that there is no extremally disconnected topological group containing a countable discrete nonclosed subset.     

Mrówka maximal almost disjoint families for uncountable cardinals

We consider generalizations of a well-known class of spaces, called by S. Mrówka, N∪R, where R is an infinite maximal almost disjoint family (MADF) of countable subsets of the natural numbers N. We denote these generalizations by ψ=ψ(κ,R) for κ?ω. Mrówka proved the interesting theorem that there exists an R such that |βψ(ω,R)?ψ(ω,R)|=1. In other words there is a unique free z-ultrafilter p₀ on the space ψ. We extend this result of Mrówka to uncountable cardinals. We show that for κ?c, Mrówka's MADF R can be used to produce a MADF M⊂ω[κ] such that |βψ(κ,M)?ψ(κ,M)|=1. For κ>c, and every M⊂ω[κ], it is always the case that |βψ(κ,M)?ψ(κ,M)|≠1, yet there exists a special free z-ultrafilter p on ψ(κ,M) retaining some of the properties of p₀. In particular both p and p₀ have a clopen local base in βψ (although βψ(κ,M) need not be zero-dimensional). A result for κ>c, that does not apply to p₀, is that for certain κ>c, p is a P-point in βψ.     

Ordinal remainders of ψ-spaces

We consider which ordinals, with the order topology, can be Stone-?ech remainders of which spaces of the form ψ(κ,M), where ω?κ is a cardinal number and M⊂ω[κ] is a maximal almost disjoint family of countable subsets of κ (MADF). The cardinality of the continuum, denoted c, and its successor cardinal, c⁺, play important roles. We show that if κ>c⁺, then no ψ(κ,M) has any ordinal as a Stone-?ech remainder. If κ?c then for every ordinal δ<κ⁺ there exists Mδ⊂ω[κ], a MADF, such that βψ(κ,Mδ)?ψ(κ,Mδ) is homeomorphic to δ+1. For κ=c⁺, βψ(κ,Mδ)?ψ(κ,Mδ) is homeomorphic to δ+1 if and only if c⁺?δ<c⁺⋅ω.     

Cardinal sequences of LCS spaces under GCH

Let C(α) denote the class of all cardinal sequences of length α associated with compact scattered spaces. Also put

C_λ(α)={f∈C(α):f(0)=λ=min[f(β):β<α]}.     

Self-duality in the class of precompact groups

A topological Abelian group G is called (strongly) self-dual if there exists a topological isomorphism Φ:G→G^∧ of G onto the dual group G^∧ (such that Φ(x)(y)=Φ(y)(x) for all x,y∈G). We prove that every countably compact self-dual Abelian group is finite. It turns out, however, that for every infinite cardinal κ with κω=κ, there exists a pseudocompact, non-compact, strongly self-dual Boolean group of cardinality κ.