Large Cardinals and Small Dowker Spaces

We prove that, if there is a model of set-theory which contains no first countable, locally compact, scattered Dowker spaces, then there is an inner model which contains a measurable cardinal. A Hausdorff space is normal if, for every pair of disjoint closed sets C and D, there is a pair of disjoint open sets, U containing C and V containing D. A (normal) space is binormal if its product with the closed unit interval I is normal. It is fair to say that the study of normality, in particular the behaviour of normality in products and the difference between normality and binormality, has played a central role in point-set topology. In 1951 Dowker [Do] introduced the notion of countable paracompactness, and proved that a normal space is binormal iff it is countably paracompact. A space is (countably) paracompact if every (countable) open cover has a locally finite open refinement, however the important point to note, as far as we are concerned, is that countable paracompactness is the difference between normality and binormality. A quick study of Dowker’s paper demonstrates just how natural the definition is—indeed, countable paracompactness is not so much a generalization of paracompactness, as one in a list of related properties which act to preserve normality-type conditions in products with a compact, metrizable factor: X × I is respectively orthocompact, perfect, δ-normal, normal, perfectly or hereditarily normal, or monotonically normal iff X is (respectively) countably metacompact [S], perfect (see [P 4.9]), countably paracompact [M], normal and countably paracompact [Do], perfectly normal [Ka] and [P 4.9], monotonically normal and semi-stratifiable [G 5.22]. Normal spaces that are not countably paracompact have become known as Dowker spaces, and it is natural to ask whether such spaces exist. In fact Dowker spaces do exist, but the example [Ru2], together with its modifications, has unsatisfyingly large cardinality and cardinal functions. This has prompted the generic definition of small Dowker spaces, i.e. ones of small size or small cardinal functions. Small Dowker spaces also exist, but, as yet, only with the help of various set-theoretic assumptions. For example: when ♣ holds, there are 1991 Mathematics Subject Classification. 03E35, 03E55, 54D15, 54D20, 54EG15.