Could not include topos theory - contents
The notion of open geometric morphism is the notion of open map for geometric morphisms between toposes.
is called open if the following equivalent conditions holds
the localic reflection of $f$ is an open map of locales;
the inverse image $f^*$ preserves first order logic, hence is a Heyting functor.
Peter Johnstone, Open maps of toposes, Manuscripta Mathematica, Volume 31, Numbers 1-3
André Joyal, Myles Tierney, An extension of the Galois theory of Grothendieck, Mem. Amer. Math. Soc. 309 (1984).