Baire Relations and Winning the Banach-Mazur Game on (X, \tau)
by
Aaron R. Todd
Baruch College, CUNY
Coauthors: M. Henriksen (Harvey Mudd College), R. Kopperman (The City College of New York, CUNY), L. Zsilinszky (U. of North Carolina, Pembroke, NC)

The existence of a winning strategy for \alpha in (a form of) the Banach-Mazur implies that a space is Baire. This property unifies many Baire category theorems and behaves well for topological operations. (Fleissner and Kunen, Fund. Math. 1978, and White, PAMS 1975) However, it is neither simple to state nor easy to use. Galvin and Telgarsky (Top&Appl. 1986) and Debs (Fund. Math. 1985) show that if a winning strategy for \alpha exists, then one exists depending on only the last move of \alpha's opponent and the previous move of \alpha.

We use their result to show that there is a winning strategy if and only if there is a Baire relation: a nonempty set R of pairs of sets with nonempty interiors such that

1. whenever (A, B) in R, A subset B,
2. whenever A subset B, have nonempty interiors, there is C subset A with (C, B) in R, and
3. \cap _n B_n =/= \emptyset whenever B₀=X, and (B_k+1, B_k) in R for each k.

Date received: June 14, 1999