For a subset of a topological space , a, interior point of is a point which has a neighbourhood in that is contained in . The union of all interior points is the interior of . It can be defined as the largest open set contained in .
In general, we have . is open if and only if , so that in particular . This makes the interior operator a co-closure operator. It also satisfies the equations and . Moreover, any co-closure operator on that preserves finite intersections must be the interior operation for some topology, namely the family consisting of fixed points of ; this gives one of many equivalent ways to define a topological space.
The interior of a subtopos of a Grothendieck topos , as well as the exterior, were defined in an exercise in SGA4: as the largest open subtopos contained in . The boundary of a subtopos is then naturally defined as the subtopos complementary to the (open) join of the exterior and interior subtoposes in the lattice of subtoposes.