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).