The stability of the Lindelöf property under the formation of products and of sums is investigated in ZF (= Zermelo-Fraenkel set theory without AC, the axiom of choice). It is • not surprising that countable summability of the Lindelöf property requires some weak choice principle, • highly surprising, however, that productivity of the Lindelöf property is guaranteed by a drastic failure of AC, ...

this paper deals with the behavior at infinity of solutions to a class of wave equations with nonlinear dampingterms defined in a semi-infinite cylinder. the spatial behavior of solutions is studied and an alternative ofphragmén-lindelöf type theorems is obtained in the results. the main point in the contribution is the use of energy method.

No convenient internal characterization of spaces that are productively Lindelöf is known. Perhaps the best general result known is Alster’s internal characterization, under the Continuum Hypothesis, of productively Lindelöf spaces which have a basis of cardinality at most א1. It turns out that topological spaces having Alster’s property are also productively weakly Lindelöf. The weakly Lindelö...