Notices Amer. Math. Sac. 51, 2004). Logically, such a "Grothendieck topos" is something like a universe of continuously variable sets. Before long, however, F.W. Lawvere and M. Tierney provided an elementary axiomatization..

We show that a morphism of locales is open if and only if all its pullbacks are skeletal in the sense of [P.T. Johnstone, Factorization theorems for geometric morphisms, II, in: Categorical Aspects of Topology and Analysis, in: Lecture Notes in Math., vol. 915, Springer-Verlag, 1982, pp. 216–233], i.e. pulling back along them preserves denseness of sublocales . This result may be viewed as the ‘dual’ of the well-known characterization of proper maps as those which are stably closed. We also (...) investigate the circumstances in which a particular sublocale, or set of sublocales, of a given locale, may be ‘declared closed’. (shrink)