Minimal covers of open manifolds with half-spaces and the proper L-S category of product spaces

Classical results about the Lusternik-Schnirelmann category of product spaces have-their analogues in the category of proper maps. By comparing the proper Lusternik-Schnirelmann category of an open manifold X with the smallest number, of closed half-spaces needed to cover X, we obtain a proper analogue of Singhof's theorem on the category of X x S-1.