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In this note we present a sketch of a negative solution of the Michael's conjecture which says that if the product Y X X is Lindelof for every hereditarily Lindelof space Y, then Y X X' is Lindelof for every hereditarily Lindelof space Y. Introduction. It is known that if Y is a hereditarily Lindelof space and X is a separable metric space or a Lindelof which is, complete in the sense of Cech, or a scattered Lindelof space, then the product Y X XW is Lindelof. The first theorem is due to E. Michael, the second one to Frolik (see [F]) and the third one to the author (see [ALJ]) (see also [T]). I think that these results were the motivation of Michael's conjecture which says that if Y X X is Lindelof for every hereditarily Lindelof space Y then Y X X' is Lindelof for every hereditarily Lindelof space Y. In this note we are going to present the sketch of the proof that the answer to the Michael's conjecture is negative. The complete proof will appear in [AL3]. Let us recall that a subset of the Cantor set is analytic if it is an image of the irrational numbers and a subset of the Cantor set is coanalytic if it is a complement of an analytic set. Examples. EXAMPLE 1. There exists Z such that for every natural number n and for every hereditarily Lindelbf space Y the product Y X Zn is Lindelbf but ZW is not. EXAMPLE 2. Let M be a coanalytic subset of the Cantor set which is not a Borel set. Then there is a space X such that, for every Lindelof space Y and every natural number n, the products Y X Xn and XW are Lindelbf but M X XW is not. It is easy to see that in order to obtain Example 1, it is enough to put Z = M X X, where M and X are from Example 2. Terminology and notation. Our topological terminology follows [E]. Let us recall that X is a P-space if every Ga-subset of X is open. The symbol N stands for the set of natural numbers and D = (0, 1) for the two-point discrete space. Greek letters are used to denote ordinal numbers, in particular X stands for the first infinite ordinal number and wI for the first uncountable ordinal number. The symbol DW stands for the Cantor set and B(io,... ,in), where ik for 0 < k ? n is an element of D, denotes the set {io} X {il} X ... X{inj X D X D X * -. If a is an ordinal Received by the editors February 4, 1982 and, in revised form, February 11, 1982 and May 10, 1982. 1980 Mathematics Subject Classification. Primary 54B10. 01983 American Mathematical Society 0002-9947/82/0000-071 8/$02.50

A note on Michael’s problem concerning the Lindelöf property in the Cartesian products