set theory - Products of Baire spaces

See:

Cohen, Paul E.
Products of Baire spaces.
Proc. Amer. Math. Soc. 55 (1976), no. 1, 119--124.

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MathSciNet review by Douglas Censer: A topological space is said to be Baire if any countable intersection of its dense open sets is dense. Assuming the continuum hypothesis (CH), J. C. Oxtoby [Fund. Math. 49 (1960/61), 157--166; MR0140638 (25 #4055); errata, MR 26, p. 1543] constructed a Baire space whose square is not Baire. The author shows that the assumption of CH is unnecessary here. The spaces considered in this paper are partially ordered $(P,leq)$ with topology generated by the initial segments. The construction of the desired Baire space with non-Baire product is based on the study of $P$ as a set of forcing conditions, as outlined by R. M. Solovay [Ann. of Math. 92 (1970), 1--56; MR0265151 MR0265151 (42 #64)].

Note that googling "products of Baire spaces" returns Cohen's article as the second hit. (The first hit is this question!)

For a positive result, see

Zsilinszky, László
Products of Baire spaces revisited.
Fund. Math. 183 (2004), no. 2, 115--121.

Excerpted from the MathSciNet review by Paul Bankston: Without extra assumptions, the product of two Baire spaces need not be Baire [see, e.g., J. C. Oxtoby, Fund. Math. 49 (1960/1961), 157--166; MR0140638 (25 #4055); P. E. Cohen, Proc. Amer. Math. Soc. 55 (1976), no. 1, 119--124; MR0401480 (53 #5307)]. This brings us to the notion of a $pi$-base; i.e., a collection of open sets such that every nonempty open set in the space contains a member of the collection. A $pi$-base each of whose members contains only countably many members of the $pi$-base is called countable-in-itself.

Oxtoby's theorem, from his 1961 paper, states that any Tikhonov (resp., finite) product of Baire spaces with countable (resp., countable-in-itself) $pi$-bases is a Baire space. The main result of the present paper is a significant strengthening of this; in particular it implies that arbitrary Tikhonov products of Baire spaces with countable-in-itself $pi$-bases are Baire spaces.

A space $X$ is called universally Kuratowski-Ulam (uK-U for short, first considered in [C. Kuratowski and S. Ulam, Fund. Math. 19 (1932), 247--251; Zbl 0005.18301]) if whenever $Y$ is a space and $E$ is a meager subset of $Xtimes Y$, the set $ Y setminus {yin Y: {xin X: (x,y)in E} text{is meager in} X} $ is meager in $Y$. The author now defines a space $X$ to be almost locally uK-U if the set ${xin X: x text{has an open uK-U neighborhood} }$ is dense in $X$.

After showing that the property of being almost locally uK-U is a proper generalization of having a countable-in-itself $pi$-base, the author proves his main theorem: only a Tikhonov (or countable box) product of Baire spaces that are almost locally uK-U is a Baire space.

The proof for the box product case is a variant of that for the Tikhonov case, and partially answers a question raised in [W. G. Fleissner, in General topology and its relations to modern analysis and algebra, IV (Proc. Fourth Prague Topological Sympos., Prague, 1976), Part B, 125--126, Soc. Czechoslovak Mathematicians and Physicists, Prague, 1977; MR0464181 (57 #4116)].