Van Douwen Research Articles

Square of countably compact groups without non-trivial convergent sequences

We show in ZFC that the existence of a countably compact Abelian group without non-trivial convergent sequences implies the existence of a countably compact group whose square is not countably compact. This improves a result obtained by van Douwen in 1980: the existence of a countably compact Boolean group without non-trivial convergent sequences implies the existence of two countably compact groups whose product is not countably compact in ZFC. Hart and van Mill showed in 1991 the existence of a countably compact group whose square is not countably compact under Martin's Axiom for countable posets. We show that the existence of such an example does not depend on some form of Martin's Axiom.

Distributivity of the algebra of regular open subsets of [formula omitted

We compare the structure of the algebras P ( ω ) / fin and A ω / Fin , where A denotes the algebra of clopen subsets of the Cantor set. We show that the distributivity number of the algebra A ω / Fin is bounded by the distributivity number of the algebra P ( ω ) / fin and by the additivity of the meager ideal on the reals. As a corollary we obtain a result of A. Dow, who showed that in the iterated Mathias model the spaces β ω ∖ ω and β R ∖ R are not co-absolute. We also show that under the assumption t = h the spaces β ω ∖ ω and β R ∖ R are co-absolute, improving on a result of E. van Douwen.

Forcing hereditarily separable compact-like group topologies on Abelian groups

Let c denote the cardinality of the continuum. Using forcing we produce a model of ZFC + CH with 2 c “arbitrarily large” and, in this model, obtain a characterization of the Abelian groups G (necessarily of size at most 2 c ) which admit: (i) a hereditarily separable group topology, (ii) a group topology making G into an S-space, (iii) a hereditarily separable group topology that is either precompact, or pseudocompact, or countably compact (and which can be made to contain no infinite compact subsets), (iv) a group topology making G into an S-space that is either precompact, or pseudocompact, or countably compact (and which also can be made without infinite compact subsets if necessary). As a by-product, we completely describe the algebraic structure of the Abelian groups of size at most 2 c which possess, at least consistently, a countably compact group topology (without infinite compact subsets, if desired). We also put to rest a 1980 problem of van Douwen about the cofinality of the size of countably compact Abelian groups.

On General Boundedness and Dominating Cardinals

For cardinals $\kappa,\lambda,\mu$ we let $\mathfrak{b}_{\kappa,\lambda,\mu}$ be the smallest size of a subset B of $^\lambda\mu$ unbounded in the sense of $\leq_\kappa$; that is, such that there is no function $f\in{}^\lambda\mu$ such that $\{\alpha<\lambda:g(\alpha)>f(\alpha)\}$ has size less than $\kappa$ for all $g\in B$. Similarly for $\mathfrak{d}_{\kappa,\lambda,\mu}$, the general dominating number, which is the smallest size of a subset B of $^\lambda\mu$ such that for every $g\in{}^\lambda\mu$ there is an $f\in B$ such that the above set has size less than $\kappa$. These cardinals are generalizations of the usual ones for $\kappa=\lambda=\mu=\omega$. When all three are the same regular cardinal, the relationships between them have been completely described by Cummings and Shelah. We also consider some variants of the functions, following van Douwen, in particular the version $\mathfrak{b}^{\uparrow}_{\kappa,\lambda,\mu}$ of $\mathfrak{b}_{\kappa,\lambda,\mu}$ in which B is required to consist of strictly increasing functions. Some of the main results of this paper are: (1) $\mathfrak{b}_{\mu,\mu,{\rm cf}\mu}\leq\mathfrak{b}_{{\rm cf}\mu,{\rm cf}\mu,{\rm cf}\mu}$; (2) for $\lambda\leq\mu$, $\mathfrak{b}^{\uparrow}_{\kappa,\lambda,\mu}$ always exists; (3) if $\mathrm{cf}\lambda= \mathrm{cf}\mu<\lambda\leq\mu$, then $\mathfrak{b}_{{\rm cf}\mu,{\rm cf}\mu,{\rm cf}\mu}= \mathfrak{b}^{\uparrow}_{\lambda,\lambda,\mu}$; (4) $\mathfrak{d}_{\omega,\mu,\mu}=\mathfrak{d}_{1,\mu,\mu}$. For background see Section 1 of the paper. Several open problems are stated.

On the cardinality of power homogeneous compacta

We prove that if X is a power homogeneous compact space then | X|⩽2 c( X)· π χ ( X) . This generalizes similar results of Arhangel'skiı̆, van Douwen and Ismail. We apply this result to get new estimates for the cardinality of (power) homogeneous compacta satisfying some special conditions.

Hereditary D-property of function spaces over compacta

It is shown that if X is compact then every subspace of Cp(X) is a D-space in the sense of E. van Douwen, which positively answers Matveev's question. A connection between the D-property and Baturov's and Grothendieck's classical theorems about function spaces over compacta is established.

𝐷-spaces and finite unions

This article is a continuation of a recent paper by the author and R. Z. Buzyakova. New results are obtained in the direction of the next natural question: how complex can a space be that is the union of two (of a finite family) “nice" subspaces? Our approach is based on the notion of a D D -space introduced by E. van Douwen and on a generalization of this notion, the notion of a D aD -space. It is proved that if a space X X is the union of a finite family of subparacompact subspaces, then X X is an a D aD -space. Under ( C H ) (CH) , it follows that if a separable normal T 1 T_1 -space X X is the union of a finite number of subparacompact subspaces, then X X is Lindelöf. It is also established that if a regular space X X is the union of a finite family of subspaces with a point-countable base, then X X is a D D -space. Finally, a certain structure theorem for unions of finite families of spaces with a point-countable base is established, and numerous corollaries are derived from it. Also, many new open problems are formulated.

Two countably compact topological groups: One of size $\aleph _\omega $ and the other of weight $\aleph _\omega $ without non-trivial convergent sequences

E. K. van Douwen asked in 1980 whether the cardinality of a countably compact group must have uncountable cofinality in ZFC. He had shown that this was true under GCH. We answer his question in the negative. V. I. Malykhin and L. B. Shapiro showed in 1985 that under GCH the weight of a pseudocompact group without non-trivial convergent sequences cannot have countable cofinality and showed that there is a forcing model in which there exists a pseudocompact group without non-trivial convergent sequences whose weight is w 1 < c. We show that it is consistent that there exists a countably compact group without non-trivial convergent sequences whose weight is N w .

ω-far points in large spaces

We prove that every normal non-compact space which is nowhere of cardinality at most c has an ω-far point. This provides a partial answer to a question of van Douwen.

On power homogeneous spaces

The notion of a Moscow space [A.V. Arhangel'skii, Comment. Math. Univ. Carolinae 24 (1983) 105–120] and a new notion of a weakly Klebanov space, introduced below, are applied to study topological properties of spaces X such that some power of X is homogeneous. Under this approach, a crucial role is played by a theorem of Y. Yajima [J. Math. Soc. Japan 36 (1984) 689–699]. Several general theorems are established that expose the general rules behind certain earlier more concrete results of E. van Douwen [Proc. Amer. Math. Soc. 69 (1978) 183–192] and V.I. Malychin [Proceedings of Leningrad International Topology Conference, Nauka, Leningrad, 1983, pp. 50–61]. In particular, it is proved that every Corson compact space X such that X τ is homogeneous, for some cardinal number τ, is first countable (Corollary 4.9). This shows why every power of the one-point compactification of an uncountable discrete space is not homogeneous [Proc. Amer. Math. Soc. 69 (1978) 183–192]. V.I. Malychin proved that ( ω 1+1) τ is not homogeneous for any τ [Proceedings of Leningrad International Topology Conference, Nauka, Leningrad, 1983, pp. 50–61]. This is covered by Corollary 4.12 below: if some power of a compact scattered space X is homogeneous, then X is countable. It is also shown that, on many occasions, βX is the only compactification of X that can be homogeneous or power homogeneous.

Normality and paranormality in product spaces

Let βω denote the Stone–Čech compactification of the countable discrete space ω. We show that if p is a point of βω⧹ ω, then all subspaces of ( ω∪{ p})× ω 1 are paranormal, where ( ω∪{ p}) is considered as a subspace of βω. This answers a van Douwen's question. Moreover we show that the existence of a paranormal non-normal subspace of ( ω+1)× ω 1 is independent of ZFC, where ω+1 is the ordinal space {0,1,2,…, ω} with the usual order topology.

Zero-dimensional spaces from linear structures

It is well known that spaces defined from a separated uniform structure with a linearly ordered base of uncountable cofinality (ω μ-metrizable spaces) are ultraparacompact and have an ortho-base, hence are non-Archimedean spaces in the sense of A. Monna. Our results concern whether certain wider classes of spaces defined from linear structures retain these properties. We construct for every regular cardinal ω μ, examples of ω μ-additive, ω μ-stratifiable spaces (i.e., ω μ-Nagata spaces) that do not have an ortho-base. We give a number of examples of linearly stratifiable spaces, one of which is related to an example of Eric K. van Douwen concerning countable box products of stratifiable spaces.

κ-normality and products of ordinals

A regular topological space is called κ- normal if any two disjoint regular closed subsets can be separated. In this paper we will show that any product of ordinals is κ-normal. In addition a generalization of a theorem of van Douwen and Vaughan will be proven and used to give an alternate proof that the product of any countable family of ordinals is κ-normal.

A solution to van Douwen's problem on Bohr topologies

We show, in ZFC, that there are two groups of the same cardinality with nonhomeomorphic Bohr topologies.

D-spaces

We use U-sticky sets to show that box products of scattered spaces of height 1 are D-spaces and to simplify the proof of van Douwen and Lutzer that a subspace of a linearly ordered space is a D-space iff no closed subset is stationary. Analogously, we show that a subspace of a product of finitely many ordinals is a D-space iff it is metacompact iff no closed subset is stationary. We examine the proof that metric spaces are D-spaces and obtain a new characterization of semistratifiability.

The Sorgenfrey line has a locally pathwise connected connectification

We answer a question of Alas, Tkacenko, Tkachuk and Wilson by constructing a connected locally pathwise connected Hausdorff space in which the Sorgenfrey line can be densely embedded. A connectification of a T2-space X is a connected Hausdorff space Y in which X can be densely embedded. In such a case Y is called a connectification of X, and X is said to be connectifiable. In 1977 Emeryk and Kulpa showed, in response to a question of van Douwen, that the Sorgenfrey line cannot be densely embedded in a connected T3-space, although it is connectifiable (see [2]). In a recent paper Alas, Tkacenko, Tkachuk and Wilson asked if the Sorgenfrey line has a locally connected connectification ([1]). The aim of this paper is to give a strongly positive answer to the above question. We refer the reader to [3] for notations and terminology not explicitly given. Recall that a collection of pairwise disjoint non-empty open subsets of a space X is called a cellular family. Theorem 1. The Sorgenfrey line has a locally pathwise connected connectification. Proof. Let S = (IR,r) be the Sorgenfrey line and let D = {dn: n E N} be a countable dense subset of S, and for every n, i E N set B(n, i) = [dn, dn + ? [ and C(n, i) = [dn + 1 v dn + B. Let Q be the subset of N3 consisting of all w = (n, m,i) with n 0, then C(n, i + k-1) E YF\; ii) if k < 0, then C(m, i k 1) E TF;; iii) the family 4D = {5X: A E A} is Hausdorff separated (i.e., if A 7 A', then there are F E F2 and F' E YA' such that F f F' = 0). Note that B((n, i) U B(m, i) E F,\. Now let H =]0, 1l[nQ ({n: n E N} U {1n E N}), and set F = Q x H. Received by the editors July 24, 1997 and, in revised form, March 25, 1999. 2000 Mathematics Subject Classification. Primary 54D35, 54D05.

Dualization of the van Douwen diagram

AbstractWe make a more systematic study of the van Douwen diagram for cardinal coefficients related to combinatorial properties of partitions of natural numbers.

Towards a Problem of E. van Douwen and A. Miller

AbstractWe discuss a problem asked by E. van Douwen and A. Miller [5] in various forcing models.

Van Douwen’s problem on 0-dimensional images of ordered compacta

A compact 0-dimensional space has a T 0 T_0 -separating rank 1 family of clopen sets iff it is the continuous image of a compact 0-dimensional linearly ordered space.

Formula omitted] does not imply [formula omitted

We prove that the cyclic monotonically normal space T of Rudin is not a J 2-space. Consequently, T has the monotone extension property but does not have D 1 ∗ or D ∗( R; + ;cch) . This answers some questions of van Douwen.