For SXS \subset X a subset of a topological space XX, a, interior point of SS is a point xSx \in S which has a neighbourhood in XX that is contained in SS. The union of all interior points is the interior S S^\circ of SS. It can be defined as the largest open set contained in SS.

In general, we have S SS^\circ \subseteq S. SS is open if and only if S =SS^\circ = S, so that in particular S =S S^{\circ\circ} = S^\circ. This makes the interior operator P(X)P(X):SS P(X) \to P(X): S \mapsto S^\circ a co-closure operator. It also satisfies the equations (ST) =S T (S \cap T)^\circ = S^\circ \cap T^\circ and X =XX^\circ = X. Moreover, any co-closure operator cc on P(X)P(X) that preserves finite intersections must be the interior operation for some topology, namely the family consisting of fixed points of cc; this gives one of many equivalent ways to define a topological space.

Compare the topological closure S¯\bar{S} and frontier S=S¯S \partial S = \bar{S} \setminus S^\circ.


The interior of a subtopos j\mathcal{E}_j of a Grothendieck topos \mathcal{E}, as well as the exterior, were defined in an exercise in SGA4: Int( j)Int(\mathcal{E}_j) as the largest open subtopos contained in j\mathcal{E}_j. 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.


Revised on August 19, 2015 04:47:11 by Thomas Holder (