Convergence-theoretic mechanisms behind product theorems

Commutation of the topologizer with products, quotientness of product maps, preservation of some properties by products, topologicity of continuous convergence, continuity of complete lattices are facets of the same quest. A new method of multifilters is used to establish (in terms of core-contour-compactness) sufficient and necessary conditions for these properties in the framework of general convergences. The relativized Antoine reflector plays here an important role. Several classical results (of Whitehead, Michael, Boehme, Cohen, Day and Kelly, Hofmann and Lawson, Schwarz and Weck, Kent and Richardson, and others) are extended or refined.