Summary: 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).