Locales, the locally fine construction and formal spaces

We investigate the connection between the spatiality of locale products and the earlier studies of the author on the locally fine coreflection of the products of uniform spaces. After giving a historical introduction and indicating the connection between spatiality and the locally fine construction, we indicate how the earlier results directly solve the first of the two open problems announced in the thesis of T. Plewe. Finally, we establish a general isomorphism between the covering monoids of the localic product of topological (completely regular) spaces and the locally fine coreflection of the corresponding product of (fine) uniform spaces. Additionally, paper relates the recent studies on formal topology and uniform spaces by showing how the transitivity of covering relations corresponds to the locally fine construction.