In this chapter, by using the duality theorem established in Chapter 3, many ideas and concepts of elementary topos theory are transferred into the context of geometric logic; these notions notably include the coHeyting algebra structure on the lattice of subtoposes of a given topos, open, closed, quasi-closed subtoposes, the dense-closed factorization of a geometric inclusion, coherent subtoposes, subtoposes with enough points, the surjection-inclusion factorization of a geometric morphism, skeletal inclusions, atoms in the lattice of subtoposes of a given topos, the Booleanization and DeMorganization of a topos. An explicit description of the Heyting operation between Grothendieck topologies on a given category and of the Grothendieck topology generated by a given collection of sieves is also obtained, as well as a number of results about the problem of ‘relativizing’ a local operator with respect to a given subtopos.