Van Douwen Research Articles

Nearly realcompact frames

A Tychonoff space X has been defined by Blair and van Douwen to be nearly realcompact if βX−υX is dense in βX−X. We extend this concept to frames. Notwithstanding the appearance of remainders in the definition – notions which do not exist, in general, in the category of frames – we obtain a conservative definition of nearly realcompact frames. We give several frame-theoretic characterisations of these types of frames. We also have a ring-theoretic characterisation which states that L is nearly realcompact precisely if the intersection of the free maximal ideals of the ring RL coincides with the intersection of its hyper-real maximal ideals. We show that P-frames and paracompact frames are nearly realcompact. Hence, Boolean frames are nearly realcompact.

Borel Tukey morphisms and combinatorial cardinal invariants of the continuum

We discuss the Borel Tukey ordering on cardinal invariants of the continuum. We observe that this ordering makes sense for a larger class of cardinals than has previously been considered. We then provide a Borel version of a large portion of van Douwen's diagram. For instance, although the usual proof of the inequality $\mathfrak p\leq\mathfrak b$ does not provide a Borel Tukey map, we show that in fact there is one. Afterwards, we revisit a result of Mildenberger concerning a generalization of the unsplitting and splitting numbers. Lastly, we show that the inclusion ordering on $\mathcal P(\omega)$ embeds into the Borel Tukey ordering on cardinal invariants.

On properties of h-homogeneous spaces with the Baire property

We describe how to assign an h-homogeneous space b ( X , k ) with a dense complete subspace and of weight k to any strongly zero-dimensional metric space X of weight ⩽ k. We investigate the properties of such spaces and obtain the conditions when b ( X 1 , k ) is homeomorphic to b ( X 2 , k ) . The h-homogeneous separable space T which is a union of B ( ω ) and a countable subspace was constructed by E. van Douwen. Similarly, the h-homogeneous separable space S which is a union of B ( ω ) and a σ-compact subspace was described by J. van Mill. These spaces are generalized for the non-separable case. We prove that if Ind X = 0 and X = G ∪ L , where G is an absolute G δ and L is of first category, then X ω is an h-homogeneous space. We consider certain cases when A × X ω is homeomorphic to X ω providing A ⊂ X ω .

Topological compactifications

We study those compactifications of a space such that every autohomeomorphism of the space can be continuously extended over the compactification. These are called H-compactifications. Van Douwen proved that there are exactly three H-compactifications of

There is a Van Douwen MAD family

We answer a long-standing question of Van Douwen by proving in ZFC that there is a MAD family of functions in ω ω that is also maximal with respect to infinite partial functions. In Section 3 we apply the idea of trace introduced in this proof to the still open question of whether analytic MAD families exist in ω ω . Using the idea of trace, we show that any analytic MAD families that may exist in ω ω must satisfy strong combinatorial constraints. We also show that it is consistent to have MAD families in ω ω that satisfy these constraints.

On some questions about KC and related spaces

Answering questions raised by O.T. Alas and R.G. Wilson, or by these two authors together with M.G. Tkachenko and V.V. Tkachuk, we show that every minimal SC space must be sequentially compact, and we produce the following examples: – a KC space which cannot be embedded in any compact KC space; – a countable KC space which does not admit any coarser compact KC topology; – a minimal Hausdorff space which is not a k-space. We also give an example of a compact KC space such that every nonempty open subset of it is dense, even if, as pointed out to us by the referee, a completely different construction carried out by E.K. van Douwen in 1993 leads to a space with the same properties.

On open ultrafilters and maximal points

In this work we expand upon the theory of open ultrafilters in the setting of regular spaces. In [E. van Douwen, Remote points, Dissertationes Math. (Rozprawy Mat.) 188 (1981) 1–45], van Douwen showed that if X is a non-feebly compact Tychonoff space with a countable π-base, then βX has a remote point. We develop a related result for the class of regular spaces which shows that in a non-feebly compact regular space X with a countable π-base, there exists a free open ultrafilter on X that is also a regular filter. Of central importance is a result of Mooney [D.D. Mooney, H-bounded sets, Topology Proc. 18 (1993) 195–207] that characterizes open ultrafilters as open filters that are saturated and disjoint-prime. Smirnov [J.M. Smirnov, Some relations on the theory of dimensions, Mat. Sb. 29 (1951) 157–172] showed that maximal completely regular filters are disjoint prime, from which it was concluded that βX is a perfect extension for a Tychonoff space X. We extend this result, and other results of Skljarenko [E.G. Skljarenko, Some questions in the theory of bicompactifications, Amer. Math. Soc. Transl. Ser. 2 58 (1966) 216–266], by showing that a maximal regular filter on any Hausdorff space is disjoint prime. Open ultrafilters are integral to the study of maximal points and lower topologies in the partial order of Hausdorff topologies on a fixed set. We show that a maximal point in a Hausdorff space cannot have a neighborhood base of feebly compact neighborhoods. One corollary is that no locally countably compact Hausdorff topology is a lower topology, which was shown previously under the additional assumption of countable tightness by Alas and Wilson [O. Alas, R. Wilson, Which topologies can have immediate successors in the lattice of T 1 -topologies? Appl. Gen. Topol. 5 (2004) 231–242]. Another is that a maximal point in a feebly compact space is not a regular point. This generalizes results of both Carlson [N. Carlson, Lower upper topologies in the Hausdorff partial order on a fixed set, Topology Appl. 154 (2007) 619–624] and Costantini [C. Costantini, On some questions about posets of topologies on a fixed set, Topology Proc. 32 (2008) 187–225].

Dually discrete spaces

A neighbourhood assignment in a space X is a family O = { O x : x ∈ X } of open subsets of X such that x ∈ O x for any x ∈ X . A set Y ⊆ X is a kernel of O if O ( Y ) = ⋃ { O x : x ∈ Y } = X . We obtain some new results concerning dually discrete spaces, being those spaces for which every neighbourhood assignment has a discrete kernel. This is a strictly larger class than the class of D-spaces of [E.K. van Douwen, W.F. Pfeffer, Some properties of the Sorgenfrey line and related spaces, Pacific J. Math. 81 (2) (1979) 371–377].

Straightening Theorem for bounded Abelian groups

We prove that every continuous function f with f ( 0 ) = 0 between two bounded Abelian groups G and H equipped with the Bohr topology coincides with a homomorphism when restricted to an infinite subset of the domain. This extends the main results of [K. Kunen, Bohr topology and partition theorems for vector spaces, Topology Appl. 90 (1998) 97–107, D. Dikranjan, S. Watson, A solution to van Douwen's problem on the Bohr topologies, J. Pure Appl. Algebra 163 (2001) 147–158]. Moreover, we give several applications and we answer a question of [B. Givens, K. Kunen, Chromatic numbers and Bohr topologies, Topology Appl. 131 (2) (2003) 189–202].

Nagata's conjecture and countably compact hulls in generic extensions

Nagata conjectured that every M -space is homeomorphic to a closed subspace of the product of a countably compact space and a metric space. Although this conjecture was refuted by Burke and van Douwen, and A. Kato, independently, but we can show that there is a c.c.c. poset P of size 2 ω such that in V P Nagata's conjecture holds for each first countable regular space from the ground model (i.e. if a first countable regular space X ∈ V is an M -space in V P then it is homeomorphic to a closed subspace of the product of a countably compact space and a metric space in V P ). By a result of Morita, it is enough to show that every first countable regular space from the ground model has a first countable countably compact extension in V P . As a corollary, we also obtain that every first countable regular space from the ground model has a maximal first countable extension in model V P .

On n -to-one continuous images of β ℕ

Eric van Douwen [5] produced a maximal crowded extremally disconnected regular space and showed that its Čech-Stone compactification is an at most two-to-one image of β ℕ. We construct for any n ≧ 3 an example of a compact crowded space Xn that is an image of β ℕ under a map all of whose fibers have either size n or n − 1. We also show that under CH this is best possible.

Classes defined by stars and neighbourhood assignments

We apply and develop an idea of E. van Douwen used to define D-spaces. Given a topological property P , the class P ∗ dual to P (with respect to neighbourhood assignments) consists of spaces X such that for any neighbourhood assignment { O x : x ∈ X } there is Y ⊂ X with Y ∈ P and ⋃ { O x : x ∈ Y } = X . We prove that the classes of compact, countably compact and pseudocompact are self-dual with respect to neighbourhood assignments. It is also established that all spaces dual to hereditarily Lindelöf spaces are Lindelöf. In the second part of this paper we study some non-trivial classes of pseudocompact spaces defined in an analogous way using stars of open covers instead of neighbourhood assignments.

On van Douwen spaces and retracts of $\beta {\mathbb{N}}$

Eric van Douwen produced in 1993 a maximal crowded extremally disconnected regular space and showed that its Stone-Čech compactification is an at most two-to-one image of $\beta {\mathbb{N}}$. We prove that there are non-homeomorphic such images. We also develop some related properties of spaces which are absolute retracts of $\beta {\mathbb{N}}$ expanding on earlier work of Balcar and Błaszczyk (1990) and Simon (1987).

A class of angelic sequential non-Fréchet–Urysohn topological groups

Fréchet–Urysohn (briefly F-U) property for topological spaces is known to be highly non-multiplicative; for instance, the square of a compact F-U space is not in general Fréchet–Urysohn [P. Simon, A compact Fréchet space whose square is not Fréchet, Comment. Math. Univ. Carolin. 21 (1980) 749–753. [27]]. Van Douwen proved that the product of a metrizable space by a Fréchet–Urysohn space may not be (even) sequential. If the second factor is a topological group this behaviour improves significantly: we have obtained (Theorem 1.6(c)) that the product of a first countable space by a F-U topological group is a F-U space. We draw some important consequences by interacting this fact with Pontryagin duality theory. The main results are the following: (1) If the dual group of a metrizable Abelian group is F-U, then it must be metrizable and locally compact. (2) Leaning on (1) we point out a big class of hemicompact sequential non-Fréchet–Urysohn groups, namely: the dual groups of metrizable separable locally quasi-convex non-locally precompact groups. The members of this class are furthermore complete, strictly angelic and locally quasi-convex. (3) Similar results are also obtained in the framework of locally convex spaces. Another class of sequential non-Fréchet–Urysohn complete topological Abelian groups very different from ours is given in [E.G. Zelenyuk, I.V. Protasov, Topologies of Abelian groups, Math. USSR Izv. 37 (2) (1991) 445–460. [32]].

Van Douwen’s diagram for dense sets of rationals

We investigate cardinal invariants related to the structure Dense ( Q ) / nwd of dense sets of rationals modulo the nowhere dense sets. We prove that s Q ≤ min { s , add ( M ) } , thus dualizing the already known r Q ≥ max { r , cof ( M ) } [B. Balcar, F. Hernández-Hernández, M. Hrušák, Combinatorics of dense subsets of the rationals, Fund. Math. 183 (2004) 59–80, Theorem 3.6]. We also show the consistency of each of h Q < s Q and h < h Q . Our results answer four questions of Balcar, Hernández and Hrušák [B. Balcar, F. Hernández-Hernández, M. Hrušák, Combinatorics of dense subsets of the rationals, Fund. Math. 183 (2004) 59–80, Questions 3.11].

The m-topology on [formula omitted] revisited

Hewitt [E. Hewitt, Rings of real-valued continuous functions, I, Trans. Amer. Math. Soc. 64 (1948) 45–99], generalizing work of E.H. Moore, defined the m-topology on C ( X ) . In his article he demonstrated that certain classes of topological spaces X can be characterized by topological properties of C m ( X ) . For example, he showed that X is pseudocompact if and only if C m ( X ) is first countable. Others have also investigated topological properties of X via properties of C m ( X ) , e.g., [G. Di Maio, L. Holá, D. Holý, R.A. McCoy, Topologies on the space of continuous functions, Topology Appl. 86 (2) (1998) 105–122] and [E. van Douwen, Nonnormality or hereditary paracompactness of some spaces of real functions, Topology Appl. 39 (1) (1991) 3–32]. We continue this practice in the second section and give some new equivalent characterizations. In the third section we prove the converse of a theorem of van Douwen [E. van Douwen, Nonnormality or hereditary paracompactness of some spaces of real functions, Topology Appl. 39 (1) (1991) 3–32] completing a characterization of when C m ( X ) is a weak P-space. In the fourth section we determine when C m ( X ) has no non-trivial convergent sequences.

On the cardinality of power homogeneous Hausdorff spaces

We prove that the cardinality of power homogeneous Hausdorff spaces X is bounded by d(X) ��(X) . This inequality improves many known results and it also solves a question by J. van Mill. We further introduce �-power homogeneity, which leads to a new proof of van Douwen's theorem.

A note on D-spaces

We introduce notions of nearly good relations and N-sticky modulo a relation as tools for proving that spaces are D-spaces. As a corollary to general results about such relations, we show that C p ( X ) is hereditarily a D-space whenever X is a Lindelöf Σ-space. This answers a question of Matveev, and improves a result of Buzyakova, who proved the same result for X compact. We also prove that if a space X is the union of finitely many D-spaces, and has countable extent, then X is linearly Lindelöf. It follows that if X is in addition countably compact, then X must be compact. We also show that Corson compact spaces are hereditarily D-spaces. These last two results answer recent questions of Arhangel'skii. Finally, we answer a question of van Douwen by showing that a perfectly normal collectionwise-normal non-paracompact space constructed by R. Pol is a D-space.

On the continuity of left translations in the LUC-compactification

Consider the continuity of left translations in the LUC-compactification G LUC of a locally compact group G. For every X ⊆ G , let κ ( X ) be the minimal cardinality of a compact covering of X in G. Let U ( G ) be the points in G LUC that are not in the closure of any X ⊆ G with κ ( X ) < κ ( G ) . We show that the points at which no left translation in U ( G ) is continuous are dense in U ( G ) . This result is a generalization of a theorem by van Douwen concerning discrete groups. We obtain a new proof for the fact that the topological center of G LUC ∖ G is empty.

Strong non-remote points

Answering to van Douwen's question, posed for Q, we construct a crowded totally non-remote point in any dense-in-itself non-compact metric space.