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By Bornemann F., Yserentant H.

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93 (1985), 195-210. Isbell, J. : First steps in descriptive theory of locales, Trans. Amer. Math. Soc. 327 (1991), 353-371; Corrections 341 (1994),467-468. Johnstone, P. : Stone Spaces, Cambridge Univ. Press, Cambridge, 1982. Joyal, A. : An extension of the Galois theory of Grothendieck, Mem. Amer. Math. Soc. 309, vol. 51, 1984. : Weak locale quotient morphisms and locally connected frames, 1. Pure Appl. Algebra 110 (1996), 101-107. : Descent theory for toposes, Bull. Soc. Math. Belgique 41 (1989), 373--391.

Pulling back along the closed sublocale A - intA(Ao) if necessary, we may assume that intA(Ao) = O. Let X = U{S :::: Ac I S /\ Ao = O} and let Y be any locally closed sublocale of A disjoint from Ac. Let X ~ A be the map induced by the various inclusions -! S -+ A and Y A be the inclusion of Y. Observe that a and {3 factor uniquely through B, say by y and 8, because (3 and each of the inclusions S -+ A used to · d'ISJomt .. fr om A m' Consl'der the map X +. Y (a,f3). ). A construct a have Images The following diagram is a pullback because (X +.

2 we have f(f-l (V) V (f-I(W) /\ = Vv (W /\Ao) which is open if Ao is. To see that (4) holds, consider any section A ~ B of f. Then H = s-I(F) is a closed sublocale which satisfies H ::: Ac and A - H ::: Ao. Conversely, if there exists closed H ::: Ac such that A - H ::: A o, then the inclusion of (f-I(H) /\ F) V (f-I(A - H) /\ U) into B is a section for f. 0 U» Recall that a map B ~ A is proper if and only if it is compact as an internal locale in the topos SheA). Two equivalent conditions are that f is stably closed, or that f is closed and f* preserves directed suprema [15].

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A basic norm equivalence for the theory of multilevel methods by Bornemann F., Yserentant H.

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