Weak products and Hausdorff locales

It is well known that one of the salient differences between the category of locales and that of spaces is that the notions of product in the two categories do not coincide: that is, if X and Y are spaces, with open-set lattices Ω(X) and Ω(Y), the locale product Ω(X) x_l Ω(Y) is in general different from Ω(X x Y).  (Note: here and throughout the paper, our notation relating to locales is taken from [5].) In this paper, we introduce a construction which goes some way towards "reconciling" this difference: specifically, we show that the assignment (Ω(X),  Ω(Y)) → Ω(X x Y) is the restriction to spatial locales of a symmetric monoidal structure defined on the whole category of locales (which we denote by (A,B) → A⊗B, and call the weak product structure), such that A⊗B is (naturally) a dense sublocale of A x₁ B for any A and B. (There is also an infinitary version of ⊗, which we shall define although we shall not investigate its properties in any great detail.)  Consideration of the extent to which ⊗ differs from the categorical product leads us to introduce anew class of "weakly spatial" locales, which forms the largest (coreflective) subcategory of Loc on which the restriction of ⊗ yields the categorical product; thanks to a recent example of Kriz and Pultr, we know that it is strictly larger than the class of spatial locales.