ON PARACOMPACTNESS IN CARTESIAN PRODUCTS AND TELGARSKY’S GAME

Abstract. We prove that if X is a paracompact space and M is a metric space such that X can be embedded in Mω1 in such a way that for each α < ω1 and the projection pα: X →Mα and any x ∈ X the mapping pα is closed at x or the set p−1α pα(x) closed and open in X then the product X×Y is paracompact for every paracompact space Y if and only if the first player of the G(DC,X) game introduced by Telgarsky has a winning strategy. One of the key problems concerning paracompactness in Cartesian products is the question: what is the characterization of the class P of all spaces whose Cartesian product with every paracompact space is paracompact. Some related problems are: Problem 1. Does X ∈ P imply that Xω is paracompact? Problem 2. Is the class P invariant with respect to closed mappings? Let us recall two conjectures which I have called Telgarsky’s conjectures: Conjecture 1. A paracompact space X belongs to P if and only if the first player in the G(DC,X) game has a winning strategy. The implication that if X is a paracompact space and the first player in the G(DC,X) game has a winning strategy then X belongs to P was proved by R.Telgarsky, see [22], Theorem 14.6. Conjecture 2. If X is a paracompact space and the second player in the G(DC,X) game has a winning strategy then X does not belong to P. This paper is a continuation of my research started in [1]- [4] and concerning paracompactness in the Cartesian products of topological spaces