Van Douwen Research Articles

Soft nodec spaces

<abstract><p>Following van Douwen, we call a soft topological space soft nodec if every soft nowhere dense subset of it is soft closed. This paper considers soft nodec spaces, which contain soft submaximal and soft door spaces. We investigate the basic properties and characterizations of soft nodec spaces. More precisely, we show that a soft nodec space can be written as a union of two disjoint soft closed soft dense (or soft open) soft nodec subspaces. Then, we study the behavior of soft nodec spaces under various operations, including the following: taking soft subspaces, soft products, soft topological sums, and images under specific soft functions with the support of appropriate counterexamples. Additionally, we show that the Krull dimension of a soft nodec soft $ T_{0} $-space is less than or equal to one. After that, we present some connections among soft nodec, soft strong nodec, and soft compact spaces. Finally, we successfully determine a condition under which the soft one-point compactification of a soft space is soft nodec if and only if the soft space is soft strong nodec.</p></abstract>

SOME REMARKS ON WASHEK PFEFFER'S CONTRIBUTIONS IN GENERAL TOPOLOGY

Washek Pfeffer's contributions in general topology are discussed, especially his paper with van Douwen on $D$-spaces and his work with Prikry on small spaces.

Disconnection in the Alexandroff duplicate

<p>It was demonstrated in [2] that the Alexandroff duplicate of the Čech-Stone compactification of the naturals is not extremally disconnected. The question was raised as to whether the Alexandroff duplicate of a non-discrete extremally disconnected space can ever be extremally disconnected. We answer this question in the affirmative; an example of van Douwen is significant. In a slightly different direction we also characterize when the Alexandroff duplicate of a space is a P-space as well as when it is an almost P-space.</p>

Reverse induction proof of D property of the countable power of the Sorgenfrey line

Answering a question of E. K. van Douwen and W. F. Pfeffer [1], we prove that the countable power of the Sorgenfrey line is a D-space. To establish this result we use a method of proof which we call reverse induction. This method allows to establish certain properties of a product \(\prod_{{i}=0}^{\infty}{X}_{i}\) by making a kind of ``reverse induction step'' from \(\prod_{{i}={n}+1}^{\infty}{X}_{i} {\rm to} \prod_{{i}={n}}^{\infty}{X}_{i}\) for an arbitrary natural n.

On metrizability and compactness of certain products without the Axiom of Choice

We prove that there exists a model of ZF (Zermelo–Fraenkel set theory without the Axiom of Choice (AC)) in which there is a compact, metrizable, non-second countable, Cantor cube. This answers in the affirmative an open question by E. Wajch (2018) [15].Furthermore, we strengthen a result in the above paper, namely “Countable products of metrizable spaces are quasi-metrizable implies van Douwen's Choice Principle”, by first observing that the above topological statement implies the weak choice principle MC(ℵ0,ℵ0), i.e. “Every countably infinite set of countably infinite sets has a multiple choice function”, and then by establishing that van Douwen's choice principle is strictly weaker than MC(ℵ0,ℵ0) in ZF. The above independence result also resolves an open problem in P. Howard and J.E. Rubin (1998) [7], and strengthens a result by P. Howard, K. Keremedis, H. Rubin, and J.E. Rubin (1998) [6].

The establishment and development of star-Lindelöf spaces

In 1983, Mumin Dai and Shôgo Ikenaga independently introduced mutually equivalent *Lindelöf spaces and ω-star spaces, later called strong 1-star-Lindelöf spaces in 1991, star-Lindelöf spaces in 1998 or star-countable spaces in 2007. After a series of work of E.K. van Douwen, M.V. Matveev and J. van Mill et al., star covering properties, the star-compactness and star-Lindelöfness being the cores, have become one of the central topics of general topology in nearly 20 years. In this paper, in clue of star-Lindelöf numbers, star covering properties and star-P properties, we review some research progress of star countable properties, introduce several unsolved problems and expound the contribution of Chinese scholars in this field.

The isomorphism class of the shift map

The shift map, denoted σ, is the self-homeomorphism of ▪ induced by the successor function n↦n+1 on ω. We prove that the isomorphism classes of σ and σ−1 cannot be separated by a Borel set in H(ω⁎), the space of all self-homeomorphisms of ω⁎ equipped with the compact-open topology.Van Douwen proved it is consistent for σ and σ−1 not to be isomorphic. Whether it is also consistent for them to be isomorphic is an open problem. The theorem stated above can be thought of as a counterpoint to van Douwen's result: while σ and σ−1 may not be isomorphic, there is no simple topological property that distinguishes them.As a relatively straightforward consequence of the main theorem, we deduce that OCA+MA implies the set of continuous images of σ fails to be Borel in H(ω⁎). (Here a “continuous image” of σ is meant in the sense of topological dynamics: any h∈H(ω⁎) such that q∘σ=h∘q for some continuous surjection q:ω⁎→ω⁎.) This contrasts starkly with a recent theorem of the author showing that under CH, the continuous images of σ form a closed subset of H(ω⁎).

Every zero-dimensional homogeneous space is strongly homogeneous under determinacy

All spaces are assumed to be separable and metrizable. We show that, assuming the Axiom of Determinacy, every zero-dimensional homogeneous space is strongly homogeneous (i.e. all its non-empty clopen subspaces are homeomorphic), with the trivial exception of locally compact spaces. In fact, we obtain a more general result on the uniqueness of zero-dimensional homogeneous spaces which generate a given Wadge class. This extends work of van Engelen (who obtained the corresponding results for Borel spaces), complements a result of van Douwen, and gives partial answers to questions of Terada and Medvedev.

Juhász’s topological generalization of Neumer’s theorem may fail in 𝖹𝖥

In set theory without the Axiom of Choice ( A C \mathsf {AC} ), we investigate the open problem of the deductive strength of Juhász’s topological generalization of Neumer’s Theorem from his paper On Neumer’s Theorem [Proc. Amer. Math. Soc. 54 (1976), 453–454]. Among other results, we show that Juhász’s Theorem is deducible from the Principle of Dependent Choices and (when restricted to the class of T 1 T_{1} spaces) implies the Axiom of Countable Multiple Choice, and hence implies van Douwen’s Countable Choice Principle, but does not imply either the full van Douwen’s Choice Principle or the axiom of choice for linearly ordered families of nonempty finite sets. Furthermore, we prove that Juhász’s Theorem (for T 1 T_{1} spaces) implies each of the following principles: “ ℵ 1 \aleph _{1} is a regular cardinal”, “every infinite set is weakly Dedekind-infinite”, and “every infinite linearly ordered set is Dedekind-infinite”. We also establish that Juhász’s Theorem for T 2 T_{2} spaces is not provable in Z F \mathsf {ZF} . In contrast to the above results, we show that Neumer’s Theorem and Juhász’s Theorem for compact T 1 T_{1} spaces are both provable in Z F \mathsf {ZF} .

Supercompact minus compact is super

According to a folklore characterization of supercompact spaces, a compact Hausdorff space is supercompact if and only if it has a binary closed k-network. This characterization suggests to call a topological space super if it has a binary closed k-network N. The binarity of N means that every linked subfamily of N is centered. Therefore, a Hausdorff space is supercompact if and only if it is super and compact. In this paper we prove that the class of super spaces contains all GO-spaces, all supercompact spaces, all metrizable spaces, and all collectionwise normal ℵ-spaces. Moreover, the class of super spaces is closed under taking Tychonoff products and discretely dense sets in Tychonoff products. The superness of metrizable spaces implies that each compact metrizable space is supercompact, which was first proved by Strok and Szymański (1975) and then reproved by Mills (1979), van Douwen (1981), and Dȩbski (1984).

Hausdorff compactifications in ZF

For a compactification αX of a Tychonoff space X, the algebra of all functions f∈C(X) that are continuously extendable over αX is denoted by Cα(X). It is shown that, in a model of ZF, it may happen that a discrete space X can have non-equivalent Hausdorff compactifications αX and γX such that Cα(X)=Cγ(X). Amorphous sets are applied to a proof that Glicksberg's theorem that βX×βY is the Čech-Stone compactification of X×Y when X×Y is a Tychonoff pseudocompact space is false in some models of ZF. It is noticed that if all Tychonoff compactifications of locally compact spaces had C⁎-embedded remainders, then van Douwen's choice principle would be satisfied. Necessary and sufficient conditions for a set of continuous bounded real functions on a Tychonoff space X to generate a compactification of X are given in ZF. A concept of a maximal functional compactification is investigated in the absence of the axiom of choice.

Countable successor ordinals as generalized ordered topological spaces

We prove the following Main Theorem: Every continuous image of a Hausdorff topological space X is a generalized ordered space if and only if X is homeomorphic to a countable successor ordinal (with the order topology). This is a generalization of E. van Douwen's result about orderable spaces.

Iterations of Stone–Čech remainders

Consider a compact space X=L0∪L1 which is a disjoint union of two Lindelöf dense subsets L0, L1, and fix this partition. Then we repeat taking remainders of Stone–Čech compactifications ω1 many times to construct a compact space Ω(X) which is “minimal” w.r.t. the property that Ω(X) admits an irreducible map π onto X such that both π−1(L0) and π−1(L1) are C⁎-embedded in Ω(X). A typical example is the closed interval [0,1]=Q∪P with the partition into the rationals Q and the irrationals P, and we can show that the resultant compact space Ω([0,1]) is not extremally disconnected, hence different from the absolute of [0,1]. We respond to one of van Douwen's questions in [2] that the length of iteration needed for Ω([0,1]) is the first uncountable ordinal ω1.

Topological difference of the iterated remainders

Repeat taking remainders of Stone–Čech compactifications of the rationals Q, the irrationals P, and the Sorgenfrey line S,Q⁎=βQ\\Q,Q⁎⁎=βQ⁎\\Q⁎,Q⁎⁎⁎,⋯;P⁎,P⁎⁎,P⁎⁎⁎,⋯;S⁎,S⁎⁎,S⁎⁎⁎,⋯. We show that all of these spaces are topologically different; the case of Q answers van Douwen's question [2].

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?

Gruff ultrafilters

We investigate the question of whether Q carries an ultrafilter generated by perfect sets (such ultrafilters were called gruff ultrafilters by van Douwen). We prove that one can (consistently) obtain an affirmative answer to this question in three different ways: by assuming a certain parametrized diamond principle, from the cardinal invariant equality d=c, and in the Random real model.

F-nodec spaces

Following Van Douwen, a topological space is said to be nodec if it satises one of the following equivalent conditions: (i) every nowhere dense subset of X, is closed; (ii) every nowhere dense subset of X, is closed discrete; (iii) every subset containing a dense open subset is open. This paper deals with a characterization of topological spaces X such that F(X) is a nodec space for some covariant functor F from the category Top to itself. T0, and FH functors are completely studied. Secondly, we characterize maps f given by a ow (X; f) in the category Set such that (X; P(f)) is nodec (resp., T0-nodec), where P(f) is a topology on X whose closed sets are precisely f-invariant sets.

(Non)connectedness and (non)homogeneity

We discuss an approach to a problem posed by A.V. Arhangel'skii and E.K. van Douwen on a possibility to present a compact space as a continuous image of a homogeneous compact space. Then we suggest some ways of proving nonhomogeneity of τ-powers of a space X using points of local connectedness (or local contractibility) and components of path connectedness of X.

A non-trivial copy of 𝛽ℕ∖ℕ

There is a copy K K of the Stone-Cech remainder, β N ∖ N = N ∗ \beta \mathbb N\setminus \mathbb N = \mathbb N^* , of the integers inside N ∗ \mathbb N^* that is not equal to D ¯ ∖ D \overline {D}\setminus D for any countable discrete D ⊂ β N D\subset \beta \mathbb N . Such a copy of N ∗ \mathbb N^* is known as a non-trivial copy of N ∗ \mathbb N^* . This answers a longstanding open problem of Eric van Douwen.

Monotone properties defined from stars of open coverings

We define some monotone properties using stars of coverings. This relates to work of J. van Mill, V. Tkachuk, R. Wilson, O. Alas, L. Junqueira, M. Matveev and others who generalized the D-space property of E. van Douwen and E. Michael. Given a property P, we call a topological space X monotonically star-P if one can assign to each open cover U a subspace s(U)⊆X with property P in such a way that st(s(U),U)=⋃{U∈U:U∩s(U)≠∅}=X and if V refines U then s(U)⊆s(V). We study monotonically star-P spaces for various compactness-like properties P.