The O functor does not preserve binary products

In Exercise 8.4.23 of the book, I said: “Exercise 8.4.21 may give you the false impression that the O functor preserves binary products. This is wrong, although an explicit counterexample seems too complicated to study here: see Johnstone (1982, 2.14).” O, here and as usual on these pages, is the open subset functor from Top to Loc.  My purpose here is to show that that is not that complicated after all.

My initial plan was to follow John Isbell’s Product spaces in locales 1981 paper (Theorem 2). The proof is only 5 lines, so that should be doable… or so I thought. But Isbell used to be very terse, and my explanation will be much longer. Read the full post.