Products of Baire Spaces

Only the usual axioms of set theory are needed to prove the existence of a Baire space whose square is not a Baire space. Assuming the continuum hypothesis (CH), Oxtoby [9] constructed a Baire space whose square is not Baire. We will show in this paper that the assumption of CH is unnecessary. Such results are greatly enhanced by Krom [5], who showed that if there is such an example, then there is also a metric example. Remarks of the referee were instrumental in making the main result of this paper an absolute one rather than one of relative consistency. In particular, the author was not aware of the forcing technique of §2. Comments on this paper by Franklin Tall were also of great help to the author. 1. Baire spaces and forcing. Suppose 9 = <[P, Si) is a partially ordered structure. 9 may be regarded as a topological space where the initial segments of 9 generate a basis. If 9 and £ are partially ordered sets, then the Cartesian product P X Q may be partially ordered pointwise to obtain a partially ordered set ?xl It is easily seen that 9 x% considered as a topological space, is homeomorphic to the product of topological spaces 9 and 2, A topological space is said to be Baire if any countable intersection of its dense open sets is dense. If the space is derived from a partially ordered set as above, then we note that any such countable intersection is necessarily open. Two elements of a partially ordered set will be called compatible if they have a common predecessor. A partially ordered structure 9 = (P, Si) will be called fine if for every p, q G P, either (l)q Si p or (2) there is an r si q which is incompatible with p. Suppose <3H is a countable standard transitive model of Zermelo-Fraenkel set theory (ZFC) and 9, S G 911 are partially ordered sets. We collect below some well-known facts. Lemma 1.0. If 9 is fine, then the following statements are equivalent. (a) 9 is Baire in <D1L (b) Whenever G is 9-generic over 91L and f G <91t[G] is an ordinal valued function with domain to, thenf G 91L. Received by the editors June 13, 1975. AMS (MOS) subject classifications (1970). Primary 02K25, 04A30; Secondary 08A10, 54B10, 54G20.