Resolvability and monotone normality

A space X is said to be 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). X is maximally resolvable if and only if it is Δ(X)-resolvable, where Δ(X) = min{|G| : G ≠ open}. We show that every crowded monotonically normal (in short: MN) space is ω-resolvable and almost μ-resolvable, where μ = min{2 ^ω , ω ₂}. On the other hand, if κ is a measurable cardinal then there is a MN space X with Δ(X) = κ such that no subspace of X is ω ₁-resolvable. Any MN space of cardinality < ℵ _ω is maximally resolvable. But from a supercompact cardinal we obtain the consistency of the existence of a MN space X with |X| = Δ(X) = ℵ _ω such that no subspace of X is ω ₂-resolvable. The preparation of this paper was supported by OTKA grant no. 61600     

Reflecting topological properties in continuous images

Given a topological property P, we study when it reflects in small continuous images, i.e., when for some infinite cardinal κ, a space X has P if and only if all its continuous images of weight less or equal to κ have P. We say that a cardinal invariant η reflects in continuous images of weight κ ⁺ if η(X) ≤ κ provided that η(Y) ≤ κ whenever Y is a continuous image of X of weight less or equal to κ ⁺. We establish that, for any infinite cardinal κ, the spread, character, pseudocharacter and Souslin number reflect in continuous images of weight κ ⁺ for arbitrary Tychonoff spaces. We also show that the tightness reflects in continuous images of weight κ ⁺ for compact spaces.     

Exponents of someN-compact spaces

For any topological spaceT, S. Mrówka has defined Exp (T) to be the smallest cardinal κ (if any such cardinals exist) such thatT can be embedded as a closed subset of the productN ^κ of κ copies ofN (the discrete space of cardinality ℵ₀). We prove that forQ, the space of the rationals with the inherited topology, Exp (Q) is equal to a certain covering number, and we show that by modifying some earlier work of ours it can be seen that it is consistent with the usual axioms of set theory including the choice that this number equal any uncountable regular cardinal less than or equal to 2^ℵ ₀. Mrówka has also defined and studied the class ℳ={κ: Exp (N _κ)=κ} whereN _κ is the discrete space of cardinality κ. It is known that the first cardinal not in ℳ must not only be inaccessible but cannot even belong to any of the first ω Mahlo classes. However, it is not known whether every cardinal below 2^ℵ ₀ is contained in ℳ. We prove that if there exists a maximal family of almost-disjoint subsets ofN of cardinality κ, then κ∈ℳ, and we then use earlier work to prove that if it is consistent that there exist cardinals which are not in the first ω Mahlo classes, then it is consistent that there exist such cardinals below 2^ℵ ₀ and that ℳ nevertheless contain all cardinals no greater than 2^ℵ ₀. Finally, we consider the relationship between ℳ and certain “large cardinals”, and we prove, for example, that if μ is any normal measure on a measurable cardinal, then μ(ℳ)=0.     

The size of epimorphic extensions

A sharp bound is given for the size of epimorphic extensions in categories of models defined over elementary logic andL _κκ where κ is strongly compact. For fragments ofL _ω1ω an example is given of a category which has a countable model with epimorphic extensions whose cardinalities approach and include the first measurable cardinal. If no measurable cardinal exists then this category has a countable model with epimorphic extensions of unbounded cardinality. This work was supported in part by the National Research Council of Canada under grant numbers A8599, A5603 and A8190. Presented by J. D. Monk.     

Finite combinations of Baire numbers

Letκ be a regular cardinal. Consider the Bair numbers of the spaces (2^θ)κ for variousθ≥κ. Letl be the number of such different Baire numbers. Models of set theory withl=1 orl=2 are known and it is also known thatl is finite. We show here that ifκ>ω, thenl could be any given finite number.     

More on countably compact,locally countable spaces

Following [5], aT ₃ spaceX is called good (splendid) if it is countably compact, locally countable (andω-fair).G(κ) (resp.S(κ)) denotes the statement that a good (resp. splendid) spaceX with |X|=κ exists. We prove here that (i) Con(ZF)→Con(ZFC+MA+2 ^ω is big+S(κ) holds unlessω=cf(κ)<κ); (ii) a supercompact cardinal implies Con(ZFC+MA+2suω>_ω+₁+┐G(ω_ω+₁); (iii) the “Chang conjecture” (ω_ω+₁),→(ω ₁,ω) implies ┐S(κ) for allκ≧k≧ω_ω; (iv) ifP addsω ₁ dominating reals toV iteratively then, in , we haveG(λ^ω) for allλ. Research supported by Hungarian National Foundation for Scientific Research grant no. 1805.     

The Lindel?f number of Cp(X)×Cp(X) for strongly zero-dimensional X

We prove that if X is a strongly zero-dimensional space, then for every locally compact second-countable space M, C _p (X, M) is a continuous image of a closed subspace of C _p (X). It follows in particular, that for strongly zero-dimensional spaces X, the Lindel?f number of C _p (X)×C _p (X) coincides with the Lindel?f number of C _p (X). We also prove that l(C _p (X ⁿ )^κ) ≤ l(C _p (X)^κ) whenever κ is an infinite cardinal and X is a strongly zero-dimensional union of at most κcompact subspaces.     

Separating ultrafilters on uncountable cardinals

A uniform ultrafilterU on κ is said to be λ-separating if distinct elements of the ultrapower never projectU to the same uniform ultrafilterV on λ. It is shown that, in the presence of CH, an ω-separating ultrafilterU on κ>ω is non-(ω, ω₁)-regular and, in fact, if κ < ℵ_ω thenU is λ-separating for all λ. Several large cardinal consequences of the existence of such an ultrafilterU are derived.     

Counting metric spaces

If κ is a cardinal number, then any class of mutually non-homeomorphic metric spaces of size κ must be a set whose cardinality cannot exceed 2 ^κ . Our main result is a vivid construction of 2 ^κ mutually non-homeomorphic complete and both path connected and locally path connected metric spaces of size κ for each cardinal number κ from continuum up. Additionally we also deal with counting problems concerning countable metric spaces and Euclidean spaces.     

A cardinal defined by a polarized partition relation

We study the first cardinalκ satisfying a partition relation defined on the set of finite sequences of smaller ordinals. We show that the fact that this cardinal is ℵ_ω is equiconsistent with the existence of a measurable cardinal. Under GCH, this cardinal must be inaccessible if it has uncountable cofinality. It is shown that the GCH assumption is necessary here.     

Chang’s conjecture for ℵω

We establish, starting from some assumptions of the order of magnitude of a huge cardinal, the consistency of (ℵ_ω+1,ℵ_ω)↠(ω₁,ω₀), as well as of some other transfer properties of the type (κ⁺,κ)↠(α⁺,α), where κ is singular.     

Projecting precipitousness

This paper is about “strong” ideals on small cardinals. It is shown that a typical property of large cardinal measures does not transfer to these ideals. More specifically, that precipitous ideals onP _ω1λ spaces may not project down to precipitous ideals on “smaller”P _ω1λ′ spaces. Also, that the existence of a presaturated ideal on the bigger space does not imply the existence of a presaturated ideal on the smaller space.     

On the weight-spectrum of a compact space

The weight-spectrumSp(w, X) of a spaceX is the set of weights of all infinite closed subspaces ofX. We prove that ifκ>ω is regular andX is compactT ₂ withω(X)≥κ then some λ withκ≤λ≤2^<κ is inSp(ω, X). Under CH this implies that the weight spectrum of a compact space can not omitω ₁, and thus solves problem 22 of [M]. Also, it is consistent with 2^ω=c being anything it can be that every countable closed setT of cardinals less thanc withω ∈ T satisfiesSp(w, X)=T for some separable compact LOTSX. This shows the independence from ZFC of a conjecture made in [AT]. Research supported by OTKA grant no. 1908.     

On the class of measurable cardinals without the axiom of choice

Using techniques of Gitik in conjunction with a large cardinal hypothesis whose consistency strength is strictly in between that of a supercompact and an almost huge cardinal, we obtain the relative consistency of the theory “ZF+⇁AC _w+κ>ω is measurable iffκ is the successor of a singular cardinal”.     

Integer-valued functions and increasing unions of first countable spaces

We determine those regular cardinals κ with the property that for each increasing κ-chain of first countable spaces there is a compatible first countable topology on the union of the chain. AssumingV=L any such κ must be weakly compact. It is relatively consistent with a supercompact cardinal that each κ>w ₁ has the property. The proofs exploit the connection with interesting families of integer-valued functions. Research of the second author supported by OTKA grant no. 1805. Research of the remaining authors partially supported by NSERC of Canada.     

SOME PROPERTIES OF STRONGLY MONOTONICALLY T2 SPACES

This paper investigates Buck＇s question about which class of spaces is strongly monotonically T2,and if other properties are combined with strongly monotonically T2,which class of spaces could be got. Based on having a cushioned pair-base space and compact strongly monotonically T2 space,some results （Theorems 1--3） are obtained.     

On the least strongly compact cardinal

We prove that under the assumption of a supercompact cardinal κ which is a limit of supercompact cardinals, for any increasing Σ₂ function φ the set {∂<κ:∂ is at least φ(∂) supercompact, is strongly compact, yet is not fully supercompact} is unbounded in κ. We then use ideas of Magidor to show that under the hypotheses of a supercompact cardinal which is a limit of supercompact cardinals it is consistent for the least strongly compact cardinal κ₀ to be at least φ(κ₀) supercompact yet not to be fully supercompact, where φ is again an increasing Σ₂ function which also meets certain other technical restrictions. The author wishes to thank Menachem Magidor for helpful conversations and suggestions in method which were used in the proof of Theorem 2.     

Baire number of the spaces of uniform ultrafilters

The Baire number is defined for a topological space without isolated points as the minimal size of the family of nowhere dense sets covering the space in question. We prove that in the case ofU(κ), the space of uniform ultrafilters over uncountable κ, the Baire number equals eitherω ₁ orω ₂, depending on the cofinality of κ. The results are connected to the collapsing of cardinals when using the quotient algebraP(κ) mod[κ]^<κ as the notion of forcing. The main portion of the present research, was done at the Center for Theoretical Study at Charles University and the Academy of Sciences.     

Some remarks on a question of D. H. Fremlin regarding ε-density

We show the relative consistency of ℵ₁ satisfying a combinatorial property considered by David Fremlin (in the question DU from his list) in certain choiceless inner models. This is demonstrated by first proving the property is true for Ramsey cardinals. In contrast, we show that in ZFC, no cardinal of uncountable cofinality can satisfy a similar, stronger property. The questions considered by D. H. Fremlin are if families of finite subsets of ω₁ satisfying a certain density condition necessarily contain all finite subsets of an infinite subset of ω₁, and specifically if this and a stronger property hold under MA + ?CH. Towards this we show that if MA + ?CH holds, then for every family ? of ℵ₁ many infinite subsets of ω₁, one can find a family ? of finite subsets of ω₁ which is dense in Fremlins sense, and does not contain all finite subsets of any set in ?. We then pose some open problems related to the question. Received: 2 June 1999 / Revised version: 2 February 2000 / Published online: 18 July 2001     

t-Sets and some algebraic properties in β S and in ℓ ∈ fty(S) *

For a large class of infinite discrete semigroups, we prove that right cancellative points in β S can have arbitrary norms or sizes. More precisely, if for x∈β S, we let ||x||= min{|A| : x ∈