How many miles to βX? II — Approximations to βX versus cofinal types of sets of metrics

Kada, Tomoyasu and Yoshinobu proved that the Stone-?ech compactification of a locally compact separable metrizable space is approximated by the collection of d-many Smirnov compactifications, where d is the dominating number. By refining the proof of this result, we will show that the collection of compatible metrics on a locally compact separable metrizable space has the same cofinal type, in the sense of Tukey relation, as the set of functions from ω to ω with respect to eventually dominating order.     

Resolvability properties via independent families

The authors give a consistent affirmative response to a question of Juhász, Soukup and Szentmiklóssy: If GCH fails, there are (many) extraresolvable, not maximally resolvable Tychonoff spaces. They show also in ZFC that for ω<λ?κ, no maximal λ-independent family of λ-partitions of κ is ω-resolvable. In topological language, that theorem translates to this: A dense, ω-resolvable subset of a space of the form (DI(λ)) is λ-resolvable.     

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.     

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.     

Lindelöf spaces which are Menger, Hurewicz, Alster, productive, or D

We discuss relationships in Lindelöf spaces among the properties “Menger”, “Hurewicz”, “Alster”, “productive”, and “D”.     

A new bound on the cardinality of homogeneous compacta

We show (in ZFC) that if X is a compact homogeneous Hausdorff space then |X|?²t(X), where t(X) denotes the tightness of X. It follows that under GCH the character and the tightness of such a space coincide.     

Amorphe Potenzen kompakter Räume

A set is amorphous, if it is not a union of two disjoint infinite subsets. The following variants of the Tychonoff product theorem are investigated in the hierarchy of weak choice principles. TA1: An amorphous power of a compactT ₂ space is compact. TA2: An amorphous power of a compactT ₂ space which as a set is wellorderable is compact. In ZF⁰TA1 is equivalent to the assertion, that amorphous sets are finite. RT is Ramsey's theorem, that every finite colouring of the set ofn-element subsets of an infinite set has an infinite homogeneous subset and PW is Rubin's axiom, that the power set of an ordinal is wellorderable. In ZF⁰RT+PW implies TA2. Since RT+PW is compatible with the existence of infinite amorphous sets, TA2 does not imply TA1 in ZF⁰. But TA2 cannot be proved in ZF⁰ alone. As an application, we prove a theorem of Stone, using a weak wellordering axiomD ₃ (a set is wellorderable, if each of its infinite subsets is structured) together with RT.

Diese Arbeit ist Teil der Habilitationsschrift des Verfassers im Fachgebiet Mathematische Analysis an der Technischen Universität Wien.     

The prime dicompletion of a di-uniformity on a plain texture

Working within a plain texture (S,S), the authors construct a completion of a dicovering uniformity υ on (S,S) in terms of prime S-filters. In case υ is separated, a separated completion is then obtained using the T₀-quotient, and it is shown that this construction produces a reflector. For a totally bounded di-uniformity it is verified that these constructions lead to dicompactifications of the uniform ditopology. A condition is given under which complementation is preserved on passing to these completions, and an example on the real texture (R,R,ρ) is presented.     

On a core concept of Arhangel'ski?

Arhangel'ski? [A.V. Arhangel'ski?, Locally compact spaces of countable core and Alexandroff compactification, Topology Appl. 154 (2007) 625-634] has introduced a weakening of σ-compactness: having a countable core, for locally compact spaces, and asked when it is equivalent to σ-compactness. We settle several problems related to that paper.     

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]}.     

A consistent σ-compact sequential space which is not hereditarily weakly Whyburn

We prove that under a=c (in particular, under Martin's Axiom) there exists a regular σ-compact sequential space which is not hereditarily weakly Whyburn. This gives a consistent solution to a question, first formulated by V.V. Tkachuk and I.V. Yashenko, and then raised again by F. Obersnel.     

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).     

A Boolean algebra and a Banach space obtained by push-out iteration

Under the assumption that c is a regular cardinal, we prove the existence and uniqueness of a Boolean algebra B of size c defined by sharing the main structural properties that P(ω)/fin has under CH and in the ℵ₂-Cohen model. We prove a similar result in the category of Banach spaces.     

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.     

Metrizable refinements of group topologies and the pseudo-intersection number

The pseudo-intersection number, denoted p, is the minimum cardinality of a family A⊆P(ω) having the strong finite intersection property but no infinite pseudo-intersection. For every countable topologizable group G, let pG denote the minimum character of a nondiscrete Hausdorff group topology on G which cannot be refined to a nondiscrete metrizable group topology. We show that pG=p.     

Different versions of a first countable space without choice

We show that two versions of a first countable topological space which are equivalent in ZFC set theory split in the absence of the Axiom of Choice AC. This answers in the negative a related question from Gutierres “What is a first countable space?”.     

Lebesgue and co-Lebesgue di-uniform texture spaces

The author introduces the notions of Lebesgue di-uniformity and co Lebesgue di-uniformity and investigates the relationship between a Lebesgue quasi uniformity on X and the corresponding Lebesgue di-uniformity on the discrete texture (X,P(X)). Finally a notion of a dual dicovering Lebesgue quasi di-uniform texture space is introduced and several properties are discussed.