Here are the details of the counterexample mentioned in my earlier e-mail, for anyone who wants to see them. Let E be the quasitopos of Frechet spaces (called subsequential spaces in my paper "On a topological topos"): all you need to know about this category is that it contains the category S of sequential spaces as a full subcategory, closed under limits (not under all colimits, but in fact all colimits that occur in the following discussion are preserved). All the action takes place inside S. Let N be the discrete space of natural numbers, and N+ its one-point compactification N \cup \{\infty\}. Let A be the space obtained from the disjoint union of two copies of N+ by identifying the two copies of \infty, and let A' be the disjoint union of N and N+. Clearly, there is a morphism A' \to A which is bijective on points (hence, both monic and epic) but not an isomorphism. However, if we regard A and A' as spaces over N+ in the obvious way, the morphism A' \to A becomes an isomorphism when we pull it back along the inclusion N \to N+, and also when we form the pushouts of A x N -------> A(') N+ | | v N since both such pushouts are isomorphic to N+. This says that, if we work in the quasitopos E/N+, the morphism A' \to A is mapped to an isomorphism in both the open subquasitopos E/N and its closed "complement". I should say that I came to consider this question as a result of a seminar talk today by Pawel Sobocinski, which raised the question of whether quasitoposes are quasi-adhesive categories. This example shows that the quasitopos of Frechet spaces is not quasi-adhesive. Peter Johnstone