Another look at the localic Tychonoff theorem.

*(English)*Zbl 0667.54009The reviewer [Fundam. Math. 113, 21-35 (1981; Zbl 0503.54006)] was the first to observe that the Tychonoff product theorem for compact locales could be proved without the axiom of choice. His proof was clumsy: in particular, it included a transfinite iteration which could have been avoided with a little ingenuity. Since then, improved versions of the proof have been given by a number of people, inluding I. Kriz and J. Vermeulen. The present paper presents what must be close to the slickest possible proof: it relies heavily on the notion of a preframe, which is a generalization of a frame in which finite joins may not exist (though directed ones do), and on an explicit description of the reflection functor from preframes to frames. (The latter was implicitly at the heart of the reviewer’s original proof, although it would be hard to detect the fact from the published version.)

Reviewer: P.T.Johnstone