Some of the combinatorics related to Michael’s problem

Abstract

We present some new methods for constructing a Michael space, a regular Lindelöf space which has a non-Lindelöf product with the space of irrationals. The central result is a combinatorial statement about the irrationals which is a necessary and sufficient condition for the existence of a certain class of Michael spaces. We also show that there are Michael spaces assuming d = cov ⁡ ( M ) \mathfrak {d}= \operatorname {cov}(\mathcal {M}) and that it is consistent with cov ⁡ ( M ) > b > d \operatorname {cov}(\mathcal {M}) > \mathfrak {b} > \mathfrak {d} that there is a Michael space. The influence of Cohen reals on Michael’s problem is discussed as well. Finally, we present an example of a Michael space of weight less than b \mathfrak {b} under the assumption that b = d = cov ⁡ ( M ) = ℵ ω + 1 \mathfrak {b} = \mathfrak {d}= \operatorname {cov} (\mathcal {M}) = \aleph _{\omega +1} (whose product with the irrationals is necessarily linearly Lindelöf).