June 16, 2006 PERFECT COMPACTA AND BASIS PROBLEMS IN TOPOLOGY

An interesting example of a compact Hausdorff space that is often presented in beginning courses in topology is the unit square [0, 1]× [0, 1] with the lexicographic order topology. The closed subspace consisting of the top and bottom edges is perfectly normal. This subspace is often called the Alexandroff double arrow space. It is also sometimes called the “split interval”, since it can be obtained by splitting each point x of the unit interval into two points x0, x1, and defining an order by declaring x0 < x1 and using the induced order of the interval otherwise. The top edge of the double arrow space minus the last point is homeomorphic to the Sorgenfrey line, as is the bottom edge minus the first point. Hence it has no countable base, so being compact, is non-metrizable. There is an obvious two-toone continuous map onto the interval. There are many other examples of non-metrizable perfectly normal compacta, if extra set-theoretic hypotheses are assumed. The most well-known is the Souslin line (compactified by adding a first and last point). Filippov[6] showed that the space obtained by “resolving” each point of a Luzin subset of the sphere S into a circle by a certain mapping is a perfectly normal locally connected non-metrizable compactum (see also Example 3.3.5 in [37]). Moreover a number of authors have obtained interesting examples under CH (or sometimes something stronger); see, e.g., Filippov and Lifanov[7], Fedorchuk[5], and Burke and Davis[3]. At some point, researchers began to wonder if there is a sense in which minor variants of the double arrow space are the only ZFC examples of perfectly normal non-metrizable compacta. A first guess was made by David Fremlin, who asked if it is consistent that every perfectly normal compact space is the continuous image of the product of the double arrow space with the unit interval. But this was too strong: Watson and Weiss[38] constructed a counterexample (which looked like the double arrow space with a countable set of isolate points added in a certain way). Finally, the following question, also due to Fremlin, became the central one: