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.