The existence of continuous weak selections and orderability-type properties in products and filter spaces

Orderability, weak orderability and the existence of continuous weak selections on filter spaces (i.e., spaces with a single non-isolated point) and their products are discussed. We prove that a closed continuous image X of a suborderable space must be hereditarily paracompact provided that its product X\times Y with some non-discrete space Y has a separately continuous weak selection.