Localic products and Till Plewe’s game

Products in the category of locales resemble, but do not coincide with products in the category of topological spaces. Till Plewe has a nice explanation to this, as I will explain in this month’s post: the localic product of two topological spaces coincides with their topological products if and only if player II has a winning strategy in a certain game, which I have already described last month. As a consequence, we will obtain that the localic product of S₀ with itself is not its topological product (a result due to Matthew de Brecht), we will retrieve that the localic product of Q and of R–Q differs from their topological product (and more generally, a result of John Isbell’s), and finally that the localic product of Q with itself differs from the topological product, and that Q is not consonant… with a much, much simpler proof than those I have ever mentioned here. The idea of that argument is due to Matthew de Brecht. Read the full post.