In mathematics, specifically in topology, the interior of a subset of a topological space is the union of all subsets of that are open in. A point that is in the interior of is an interior point of. The interior of is the complement of the closure of the complement of. In this sense interior and closure are dual notions. The exterior of a set is the complement of the closure of ; it consists of the points that are in neither the set nor its boundary. The interior, boundary, and exterior of a subset together partition the whole space into three blocks. The interior and exterior are always open while the boundary is always closed. Sets with empty interior have been called boundary sets.
Definitions
Interior point
If is a subset of a Euclidean space, then is an interior point of if there exists an open ball centered at which is completely contained in. This definition generalizes to any subset of a metric space with metric : is an interior point of if there exists, such that is in whenever the distance. This definition generalises to topological spaces by replacing "open ball" with "open set". Let be a subset of a topological space. Then is an interior point of if is contained in an open subset of which is completely contained in.
Interior of a set
The interior of a subset of a topological space, denoted by or, can be defined in any of the following equivalent ways:
If one considers on the topology in which every set is open, then.
If one considers on the topology in which the only open sets are the empty set and itself, then is the empty set.
These examples show that the interior of a set depends upon the topology of the underlying space. The last two examples are special cases of the following.
Note that these properties are also satisfied if "interior", "subset", "union", "contained in", "largest" and "open" are replaced by "closure", "superset", "intersection", "which contains", "smallest", and "closed", respectively. For more on this matter, see interior operator below.
Interior operator
The interior operatoro is dual to the closure operator—, in the sense that and also where is the topological space containing, and the backslash refers to the set-theoretic difference. Therefore, the abstract theory of closure operators and the Kuratowski closure axioms can be easily translated into the language of interior operators, by replacing sets with their complements. In general, the interior operator does not commute with unions. However, in a complete metric space the following result does hold:
Exterior of a set
The exterior of a subset of a topological space, denoted or, is the interior of its relative complement. Alternatively, it can be defined as, the complement of the closure of. Many properties follow in a straightforward way from those of the interior operator, such as the following.
is the union of all open sets that are disjoint with.
is the largest open set that is disjoint with.
If, then is a superset of.
Unlike the interior operator, ext is not idempotent, but the following holds:
is a superset of.
Interior-disjoint shapes
Two shapes and are called interior-disjoint if the intersection of their interiors is empty. Interior-disjoint shapes may or may not intersect in their boundary.
