Small Dowker Spaces

Open Problems in Topology II – Edited by E. Pearl © 2007 Elsevier B.V. All rights reserved Small Dowker spaces Paul J. Szeptycki A normal space whose product with the closed unit interval I is not normal is called a Dowker space. The spaces are named for Hugh Dowker who proved a number of characterizations of the class of spaces for which X × I is normal. One of his motivations for studying this class of spaces was the study of the so- called insertion property: X has the insertion property whenever f, g : X → R such that f < g, g lower semi-continuous and f upper semi-continuous, there is a continuous h: X → R such that f < h < g. Dowker proved the following characterization ([17]). Theorem. For a normal space, the following are equivalent (1) X is countably paracompact. (2) X × Y is normal for all infinite, compact metric spaces Y . (3) X × Y is normal for some infinite compact metric space Y . (4) X has the insertion property. The homotopy extension property also fueled interest in normality of products X×I (later Starbird [46] and Morita [34] showed that normality of X was sufficient to for the homotopy extension property). Dowker was the first to raise the question when he asked (in a footnote to [17]): “It would be interesting to have an example of a normal Hausdorff space that is not countably paracompact.” Indeed. The connection between normality and countable paracompactness is quite subtle and constructing a normal not countably paracompact space has turned out to be a difficult and deep problem. The first few examples of Dowker spaces were all constructed by M.E. Rudin. In [39], assuming the existence of Suslin tree, she constructed a locally countable realcompact Dowker space of size ℵ1. At the time Suslin’s hypothesis was still open, but, nonetheless, this was the first construction of any kind of Dowker space. Moreover, it established a blueprint for the construction of many Dowker spaces to come. The construction is also fairly flexible and can be modified to obtain locally compact, first countable examples assuming the existence of a Suslin tree. Rudin’s next example was the first, and for many years only, ZFC Dowker space [40]. An easy to describe subspace of the box product {ωn+1 : 0 < n < ω}, it is strongly collectionwise normal ([23]) and orthocompact ([24]), but it has very ℵ0 few other nice properties. For example, the cardinality of the space is ℵω and all other natural local and global cardinal functions on this space are equally large. Although this example answered Dowker’s original question, instead of closing the book on the Dowker space problem, it raised what has come to be known as the ‘small Dowker space problem.’ A small Dowker space means any Dowker space with some small local or global cardinal invariant. Here small is usually interpreted The author acknowledges support from NSERC grant 238944. 233 234 §27. Szeptycki, Small Dowker spaces as countable, ω1 or less than or equal continuum. Each of the following problems was raised in one of Rudin’s early papers and they are still open in ZFC: 487? Problem 1. Is there a Dowker space of size ℵ1? 488? Problem 2. Is there a first countable Dowker space? 489? Problem 3. Is there a separable Dowker space? 490? Problem 4. Is there a locally compact Dowker space? Since most of the consistent examples of Dowker spaces of size ω1 are locally countable it is natural to ask whether such spaces exist in ZFC: 491? Problem 5. Is there a locally countable Dowker space? Rudin’s ZFC example and the questions it raised, led to a flurry of activity fueled by the set-theoretic developments of the 1970s and 1980s. During that time many very special consistent examples were constructed under a variety of different assumptions. In addition, a few other ZFC examples were obtained by modifying Rudin’s example ([16, 19, 25, 29, 30]). No truly new ZFC example of any kind of Dowker space, small or large, was constructed until the celebrated example of Balogh published more than 20 years after Rudin’s example [3]. Balogh’s example is of size continuum and is the first ZFC example of a small Dowker space. While the example itself is, in the words of Jerry Vaughan “a milestone in set- theoretic topology” [50], more remarkable is the main technique invented for the construction. Balogh developed an elementary submodel technique based on ideas of M.E. Rudin. This technique is quite flexible (at least for Balogh) who previously used it to construct Q-set spaces, and later to solve Morita’s conjectures [4, 6]. He also employed it to construct a few other more specialized small Dowker spaces in ZFC: a screenable example [5] and a hereditarily collectionwise normal, hereditarily meta-Lindelöf, hereditarily realcompact example [8]. Perhaps the best reference for somebody wishing to learn the technique is the posthumous [7]. While all the examples are of size 2ℵ0 , none is first countable, locally compact, locally countable or separable. So although Balogh’s examples are small Dowker spaces, they do not settle any of the main Questions 1–5 in ZFC. Except for a few consistent examples constructed quite recently (that I will mention below) that is pretty much the short history of the small Dowker space problem to date. Returning to the main Questions 1–5, it is remarkable how few independence results have been obtained. There are a handful of independence results showing that certain constructions cannot yield ZFC examples, but there are very few quotable results. For example, PFA implies there is no hereditarily separable Dowker space (since all such examples are hereditarily Lindelof assuming PFA [49]). Slightly more technical, but also important is Balogh’s result (see [2]) that under MA + ¬CH there is no locally countable locally compact submetrizable Dowker space of size ℵ1 (a space is submetrizable if it has a weaker metric topology). Note that an example announced in [28] assuming ♦ has all these properties. However, none of the modifications of Rudin’s example, nor Balogh’s examples (see the discussion in [37]) are submetrizable. 235 Problem 6. Is there a submetrizable Dowker space? 492? Illustrating how weak our independence results seem, none rules out a single ZFC example giving a positive answer to Problems 1–5 simultaneously: Problem 7. Is there a ZFC example of a locally compact, locally countable (hence 493? first countable) separable Dowker space of size ℵ1? This gives some support to conjecture positive answers to the main questions. Further evidence is given by an example of Chris Good [22]: assuming that no inner model contains a measurable cardinal, there is a locally compact, locally countable (hence first countable) zero-dimensional, collectionwise normal, σ-discrete Dowker space. Therefore, the consistency of no first countable Dowker space, or no locally compact Dowker space would require at least a measurable cardinal. Of course, Good’s example is large in cardinality and sheds no light on the existence of a Dowker spaces of size ℵ1, or a separable example. Part of the difficulty in showing the consistency of no Dowker space of size ℵ1 is that we have no natural set-theoretic formulation of the problem (compare with the S-space and L-space problems). This shortcoming was touched upon by the following question of Rudin [42]: Problem 8. Find a purely set-theoretic translation of the assertion: “There is a 494? Dowker space of size ℵ1.” Rudin also asked whether the existence of a Dowker space of size ℵ1 implies the existence of a Souslin tree [42]. We now know the answer to this question is ‘no’ (there are models of CH without Suslin trees and CH implies the existence of some very nice Dowker spaces of size ℵ1, e.g., [28]). However, it is possible to revive Rudin’s question if we strengthen the hypotheses: (local compactness may be a particularly important assumption, see the discussion on locally compact Dowker spaces below). A positive answer to the following version of Rudin’s question is possible (but would be surprising): Problem 9. Does the existence of (some kind of) locally compact (maybe add 495? locally countable, realcompact, σ-discrete) Dowker space of size ℵ1 imply the existence of a Suslin tree? While there are many sufficient conditions established for the existence of certain small Dowker spaces, establishing any kind of necessary conditions would seem to require some new ideas. It is also not known if either MA + ¬CH or PFA say something about Dowker spaces of size ℵ1: Problem 10. Does MA + ¬CH or PFA imply that there are no Dowker spaces of 496? size ω1? Barring a ZFC example of size ℵ1, it is natural to ask what is the minimum cardinality of a Dowker space. For the purposes of brevity, let us denote this car- ω dinal by D. Rudin’s example is of cardinality ℵω and Balogh’s example is of size 236 §27. Szeptycki, Small Dowker spaces ℵ0 ℵ0 continuum, and it is consistent that 2 = ℵω . Kojman and Shelah constructed Q a subspace of Rudin’s ZFC example using a scale in n∈ω ωn of cardinality ℵω+1 (such scales exist in ZFC [45]). Later Kojman and Lubitch constructed a similar example with even nicer properties [29]. These examples gives the current best absolute bound for D. 497? Problem 11 (Kojman and Shelah; Watson). Is it consistent that every Dowker space is of cardinality ≥ ℵω+1? Balogh’s example of size continuum shows that D may be considered a cardinal invariant of the continuum.

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