Ultrafilters related to the Michael space problem

A Lindelöf Topological space is Michael if it has non-Lindelöf product with the space of the irrational numbers. These kind of spaces were introduced by Ernest Michael in 1963 and it is still unknown if one can be constructed in ZFC. We will introduce the notion of Michael ultrafilter and we will construct a Michael space from this ultrafilter. We will also discuss the existence of this kind of ultrafilter in some classical models of set theory.