On the products of weakly Lindelöf spaces

Abstract

The aim of this note is to show, without using any special set-theoretic assumptions, that the product of two (weakly) Lindelo'f spaces is not necessarily weakly Lindelof. In (1) M. Ulmer has constructed two weakly Lindelof spaces whose prod- uct is not so; in his construction, the assumption 2 =2 was essentially employed. In this short note we shall provide another such example (where the factors are even Lindelof), in the construction of which no additional set-theoretic assumption is used. To start with, we shall deal with some properties of the topologies on linearly ordered sets which may be interesting in themselves. We recall that, given a cardinal number a, X is (weakly) a-Lindelof if every open cover of X has a subcover (weak subcover, i.e. a subfamily whose union is dense in X) of cardinality < a. Let (/?,-<) be a linearly ordered set. We shall denote by R + and R~, respectively, the spaces on R tot which the half-open intervals of the form (x, y) and (x, y), respectively, form an open basis.