Pointless uniformities. I: Complete regularity. II: (Dia)metrization.

*(English)*Zbl 0543.54023
Commentat. Math. Univ. Carol. 25, 91-104 (1984); 25, 105-120 (1984).

The author characterizes completely regular locales by the existence of compatible uniformities or of suitable real-valued functions called diameters. The main results of the second part are that each uniformity on a locale is induced by a family of diameters; if it has a countable basis, by a single diameter. There are also some results on some not- quite uniformities called weak uniformities. (This conflicts with the well-established terminology of weak uniformities and weak topologies induced by families of functionals.)

Reviewer: J.R.Isbell