ZFC Example Research Articles

DIAMOND ON LADDER SYSTEMS AND COUNTABLY METACOMPACT TOPOLOGICAL SPACES

Abstract The property of countable metacompactness of a topological space gets its importance from Dowker’s 1951 theorem that the product of a normal space X with the unit interval $[0,1]$ is again normal iff X is countably metacompact. In a recent paper, Leiderman and Szeptycki studied $\Delta $ -spaces, which is a superclass of the class of countably metacompact spaces. They proved that a single Cohen real introduces a ladder system $ L$ over the first uncountable cardinal for which the corresponding space $X_{ L}$ is not a $\Delta $ -space, and asked whether there is a ZFC example of a ladder system $ L$ over some cardinal $\kappa $ for which $X_{ L}$ is not countably metacompact, in particular, not a $\Delta $ -space. We prove that an affirmative answer holds for the cardinal $\kappa =\operatorname {\mathrm {cf}}(\beth _{\omega +1})$ . Assuming $\beth _\omega =\aleph _\omega $ , we get an example at a much lower cardinal, namely $\kappa =2^{2^{2^{\aleph _0}}}$ , and our ladder system L is moreover $\omega $ -bounded.

Lindelöf domination versus $$\omega$$-domination of discrete subsets

A discrete subset is said to be Lindelof dominated (respectively, $$\omega$$-dominated) if it is contained in the closure of a Lindelof (respectively, a countable) subspace. We continue the study of spaces begun in [1] in which every discrete subset is Lindelof dominated (respectively, $$\omega$$-dominated). We generalize results of [1] and [15] concerning perfect Hausdorff spaces and give a ZFC example of a perfect space in which all discrete subsets are Lindelof dominated but not $$\omega$$-dominated.

The structure of locally compact normal spaces: Some quasi-perfect preimages

A quasi-perfect map is a continuous, closed function such that the preimage of every point is countably compact. An ambitious old problem due to van Douwen [1] is whether every first countable regular space of cardinality ≤c is a quasi-perfect image of a locally compact space. Here we construct locally compact, normal, quasi-perfect preimages for all stationary, co-stationary subsets of ω1. These subsets are ω1-compact but not σ-countably compact. Quasi-perfect preimages preserve these properties, and the preimages constructed here are all of cardinality b. This provides a new upper bound for the following problem:What is the least cardinality of a ZFC example of a locally compact,ω1-compact space that is not σ-countably compact?

On the problem of compact totally disconnected reflection of nonmetrizability

We construct a ZFC example of a nonmetrizable compact space K such that every totally disconnected closed subspace L⊆K is metrizable. In fact, the construction can be arranged so that every nonmetrizable compact subspace may be of fixed big dimension. Then we focus on the problem of whether a nonmetrizable compact space K must have a closed subspace with a nonmetrizable totally disconnected continuous image. This question has several links with the structure of the Banach space C(K), for example, by Holsztyński's theorem, if K is a counterexample, then C(K) contains no isometric copy of a nonseparable Banach space C(L) for L totally disconnected. We show that in the literature there are diverse consistent counterexamples, most of them eliminated by Martin's axiom and the negation of the continuum hypothesis, but some consistent with it. We analyze the above problem for a particular class of spaces. OCA however, implies the nonexistence of any counterexample in this class but the existence of another absolute example remains open.

A weak$^*$ separable $C(K)^*$ space whose unit ball is not weak$^*$ separable

We provide a ZFC example of a compact space K such that C(K)* is w*-separable but its closed unit ball is not w*-separable. All previous examples of such kind had been constructed under CH. We also discuss the measurability of the supremum norm on that C(K) equipped with its weak Baire sigma-algebra.

Maximal pseudocompact spaces and the Preiss-Simon property

Abstract We study maximal pseudocompact spaces calling them also MP-spaces. We show that the product of a maximal pseudocompact space and a countable compact space is maximal pseudocompact. If X is hereditarily maximal pseudocompact then X × Y is hereditarily maximal pseudocompact for any first countable compact space Y. It turns out that hereditary maximal pseudocompactness coincides with the Preiss-Simon property in countably compact spaces. In compact spaces, hereditary MP-property is invariant under continuous images while this is not true for the class of countably compact spaces. We prove that every Fréchet-Urysohn compact space is homeomorphic to a retract of a compact MP-space. We also give a ZFC example of a Fréchet-Urysohn compact space which is not maximal pseudocompact. Therefore maximal pseudocompactness is not preserved by continuous images in the class of compact spaces.

Integration in Hilbert generated Banach spaces

We prove that McShane and Pettis integrability are equivalent for functions taking values in a subspace of a Hilbert generated Banach space. This generalizes simultaneously all previous results on such equivalence. On the other hand, for any super-reflexive generated Banach space having density character greater than or equal to the continuum, we show that Birkhoff integrability lies strictly between Bochner and McShane integrability. Finally, we give a ZFC example of a scalarly null Banach space-valued function (defined on a Radon probability space) which is not McShane integrable.

Entire functions mapping uncountable dense sets of reals onto each other monotonically

When A A and B B are countable dense subsets of R \mathbb {R} , it is a well-known result of Cantor that A A and B B are order-isomorphic. A theorem of K.F. Barth and W.J. Schneider states that the order-isomorphism can be taken to be very smooth, in fact the restriction to R \mathbb {R} of an entire function. J.E. Baumgartner showed that consistently 2 ℵ 0 > ℵ 1 2^{\aleph _0}>\aleph _1 and any two subsets of R \mathbb {R} having ℵ 1 \aleph _1 points in every interval are order-isomorphic. However, U. Abraham, M. Rubin and S. Shelah produced a ZFC example of two such sets for which the order-isomorphism cannot be taken to be smooth. A useful variant of Baumgartner’s result for second category sets was established by S. Shelah. He showed that it is consistent that 2 ℵ 0 > ℵ 1 2^{\aleph _0}>\aleph _1 and second category sets of cardinality ℵ 1 \aleph _1 exist while any two sets of cardinality ℵ 1 \aleph _1 which have second category intersection with every interval are order-isomorphic. In this paper, we show that the order-isomorphism in Shelah’s theorem can be taken to be the restriction to R \mathbb {R} of an entire function. Moreover, using an approximation theorem of L. Hoischen, we show that given a nonnegative integer n n , a nondecreasing surjection g : R → R g\colon \mathbb {R}\to \mathbb {R} of class C n C^n and a positive continuous function ϵ : R → R \epsilon \colon \mathbb {R}\to \mathbb {R} , we may choose the order-isomorphism f f so that for all i = 0 , 1 , … , n i=0,1,\dots ,n and for all x ∈ R x\in \mathbb {R} , | D i f ( x ) − D i g ( x ) | > ϵ ( x ) |D^if(x)-D^ig(x)|>\epsilon (x) .

Strictly positive measures on Boolean algebras

AbstractWe investigate strictly positive finitely additive measures on Boolean algebras and strictly positive Radon measures on compact zerodimensional spaces. The motivation is to find a combinatorial characterisation of Boolean algebras which carry a strictly positive finitely additive finite measure with some additional properties, such as separability or nonatomicity. A possible consistent characterisation for an algebra to carry a separable strictly positive measure was suggested by Talagrand in 1980, which is that the Stone space K of the algebra satisfies that its space M(K) of measures is weakly separable, equivalently that C(K) embeds into l∞. We show that there is a ZFC example of a Boolean algebra (so of a compact space) which satisfies this condition and does not support a separable strictly positive measure. However, we use this property as a tool in a proof which shows that under MA + ¬ CH every atomless ccc Boolean algebra of size < c carries a nonatomic strictly positive measure. Examples are given to show that this result does not hold in ZFC. Finally, we obtain a characterisation of Boolean algebras that carry a strictly positive nonatomic measure in terms of a chain condition, and we draw the conclusion that under MA + ¬ CH every atomless ccc Boolean algebra satisfies this stronger chain condition.

Covering by discrete and closed discrete sets

Say that a cardinal number κ is small relative to the space X if κ < Δ ( X ) , where Δ ( X ) is the least cardinality of a non-empty open set in X. We prove that no Baire metric space can be covered by a small number of discrete sets, and give some generalizations. We show a ZFC example of a regular Baire σ-space and a consistent example of a normal Baire Moore space which can be covered by a small number of discrete sets. We finish with some remarks on linearly ordered spaces.

An L space with a d-separable square

In this note I will show that an inessential modification of the ZFC example of a hereditarily Lindelöf nonseparable space which was constructed in [J.T. Moore, A solution to the L space problem, J. Amer. Math. Soc. 19 (3) (2006) 717–736] has the property that its square contains a σ-discrete dense set.

Pseudocompactness of hyperspaces

We consider the following question of Ginsburg: Is there any relationship between the pseudocompactness of X ω and that of the hyperspace 2 X ? We do that first in the context of Mrówka–Isbell spaces Ψ ( A ) associated with a maximal almost disjoint (MAD) family A on ω answering a question of J. Cao and T. Nogura. The space Ψ ( A ) ω is pseudocompact for every MAD family A . We show that (1) ( p = c ) 2 Ψ ( A ) is pseudocompact for every MAD family A . (2) ( h < c ) There is a MAD family A such that 2 Ψ ( A ) is not pseudocompact. We also construct a ZFC example of a space X such that X ω is pseudocompact, yet 2 X is not.

First countable spaces without point-countable π-bases

We answer several questions of V. Tkaycuk from (Point- countable �-bases in first countable and similar spaces, Fund. Math. 186 (2005), pp. 55-69.) by showing that (1) there is a ZFC example of a first countable, 0-dimensional Hausdorff space with no point-countable �-base (in fact, the order of any �-base of the space is at least ℵ!); (2) if there is a �-Suslin line then there is a first countable GO space of cardinality � + in which the order of any �-base is at least �; (3) it is consistent to have a first countable, hereditarily Lindelof regular space having uncountable �-weight and !1 as a caliber (of course, such a space cannot have a point-countable �- base).

First countable, countably compact spaces and the continuum hypothesis

We build a model of ZFC+CH in which every first countable, countably compact space is either compact or contains a homeomorphic copy ofω1\omega _1with the order topology. The majority of the paper consists of developing forcing technology that allows us to conclude that our iteration adds no reals. Our results generalize Saharon Shelah’s iteration theorems appearing in Chapters V and VIII ofProper and improper forcing(1998), as well as Eisworth and Roitman’s (1999) iteration theorem. We close the paper with a ZFC example (constructed using Shelah’s club–guessing sequences) that shows similar results do not hold for closed pre–images ofω2\omega _2.

Perfectly meager sets and universally null sets

We will show that there is no ZFC example of a set distinguishing between universally null and perfectly meager sets.

Pytkeev spaces and sequential extensions

In this paper we construct, in response to a question of Malykhin and Tironi, a ZFC example of a perfectly normal Pytkeev space which has no sequential extensions.

Products of Michael spaces and completely metrizable spaces

For disjoint subsets A,C of [0, 1] the Michael space M(A,C) = A ∪ C has the topology obtained by isolating the points in C and letting the points in A retain the neighborhoods inherited from [0, 1]. We study normality of the product of Michael spaces with complete metric spaces. There is a ZFC example of a Lindelof Michael space M(A,C), of minimal weight א1, with M(A,C) × B(א0) Lindelof but with M(A,C) × B(א1) not normal. (B(אα) denotes the countable product of a discrete space of cardinality אα.) If M(A) denotes M(A, [0, 1]rA), the normality of M(A)×B(א0) implies the normality of M(A)× S for any complete metric space S (of arbitrary weight). However, the statement “M(A,C)×B(א1) normal implies M(A,C)×B(א2) normal” is axiom sensitive.

Two examples concerning almost continuous functions

In this note we will construct, under the assumption that union of less than continuum many meager subsets of R is meager in R, an additive connectivity function f:R→R with Cantor intermediate value property which is not almost continuous. This gives a partial answer to a question of Banaszewski (1997). (See also Question 5.5 of Gibson and Natkaniec (1996–97).) We will also show that every extendable function g:R→R with a dense graph satisfies the following stronger version of the SCIVP property: for every a<b and every perfect set K between g(a) and g(b) there is a perfect set C⊂(a,b) such that g[C]⊂K and g↾C is continuous strictly increasing. This property is used to construct a ZFC example of an additive almost continuous function f:R→R which has the strong Cantor intermediate value property but is not extendable. This answers a question of Rosen (1997–98). This also generalizes Rosen's result (1997–98) that a similar (but not additive) function exists under the assumption of the Continuum Hypothesis, and gives a full answer to Question 3.11 of Gibson and Natkaniec (1996–1997).

Some topological groups with and some without suitable sets

If a discrete subset S of a topological group G with identity 1 generates a dense subgroup of G and S∪{1} is closed in G, then S is called a suitable set for G. We construct in ZFC a Lindelöf topological group L such that t(L)·ψ(L)≤ℵ0 and L does not have a suitable set. We also give a ZFC example of a countably compact topological group H with no suitable set; in addition, the closure of every countable subset of H is compact. It is proved that a non-pseudocompact topological group with a dense strictly σ-discrete subset has a closed suitable set. This implies, in particular, that a free (Abelian) topological group on a metrizable space has a closed suitable set.

Maps of Ostaszewski and related spaces

Under ♦ there is an Ostaszewski space which is retractive, homeomorphic to every uncountable closed subspace, and homeomorphic to every locally countable regular Hausdorff uncountable continuous image, and also an Ostaszewski space with none of these properties. There is a ZFC example of a thin-tall locally compact scattered space for which each nonempty Cantor-Bendixson remainder is a retract. Attention is also paid to robustness under various types of forcing.