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 $\mathfrak {d}= \operatorname {cov}(\mathcal {M})$ and that it is consistent with $\operatorname {cov}(\mathcal {M}) < \mathfrak {b} < \mathfrak {d}$ 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 $\mathfrak {b}$ under the assumption that $\mathfrak {b} = \mathfrak {d}= \operatorname {cov} (\mathcal {M}) = \aleph _{\omega +1}$ (whose product with the irrationals is necessarily linearly Lindelöf).