A constructive proof of the Tychonoff’s theorem for locales. (English) Zbl 0661.54027

Kříž, Igor

Commentat. Math. Univ. Carol. 26, 619-630 (1985).

Tikhonov’s theorem stating that a product of compact spaces is compact is well known to be equivalent to the axiom of choice. A surprising result was obtained by P. T. Johnstone in Fundam. Math. 113, 21-35 (1981; Zbl 0503.54006): If we consider compact locales, the analogon of Tikhonov’s theorem can be proved without the axiom of choice. This is particularly interesting in connection with the fact that compact locales are always spatial, i.e. open-sets lattices of classical topological spaces (thus, the use of AC is localized in the formation of points, not in the preservation of the compactness property). Johnstone’s proof contains a non-constructiv element, namely the axiom of replacement. He formulated the problem whether one can get rid of this, too (for the special case of the locally compact locales he presented a positive answer himself). In this article, this problem is solved in the affirmative in full generality. The procedure is based on a new description of the product of locales, considerably more constructive as compared with the usually used ones.

MSC:

54D30	Compactness
54B10	Product spaces in general topology
54F05	Linearly ordered topological spaces, generalized ordered spaces, and partially ordered spaces

Keywords:

frame; axiom of choice; compact locales; axiom of replacement

Citations:

Zbl 0503.54006