Products of general Menger spaces

We study products of general topological spaces with Menger's covering property, and its refinements based on filters and semifilters. To this end, we extend the projection method from the classic real line topology to the Michael topology. Among other results, we prove that, assuming \CH{}, every productively Lindel\"of space is productively Menger, and every productively Menger space is productively Hurewicz. None of these implications is reversible.