In topology

Topology

Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...

, the exterior of a subset S of a topological space

Topological space

Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...

 X is the union

Union (set theory)

In set theory, the union of a collection of sets is the set of all distinct elements in the collection. The union of a collection of sets S_1, S_2, S_3, \dots , S_n\,\! gives a set S_1 \cup S_2 \cup S_3 \cup \dots \cup S_n.- Definition :...

 of all open set

Open set

The concept of an open set is fundamental to many areas of mathematics, especially point-set topology and metric topology. Intuitively speaking, a set U is open if any point x in U can be "moved" a small amount in any direction and still be in the set U...

s of X which are disjoint from S. It is itself an open set and is disjoint from S. The exterior of S is denoted by

ext S

or

S^e.

Equivalent definitions

The exterior is equal to X \ S^—, the complement

Complement (set theory)

In set theory, a complement of a set A refers to things not in , A. The relative complement of A with respect to a set B, is the set of elements in B but not in A...

 of the topological closure of S and to the interior

Interior (topology)

In mathematics, specifically in topology, the interior of a set S of points of a topological space consists of all points of S that do not belong to the boundary of S. A point that is in the interior of S is an interior point of S....

of the complement of S in X.

Properties

Many properties follow in a straightforward way from those of the interior operator, such as the following.

• ext(S) is an open set that is disjoint with S.
• ext(S) is the union of all open sets that are disjoint with S.
• ext(S) is the largest open set that is disjoint with S.
• If S is a subset of T, then ext(S) is a superset of ext(T).

Unlike the interior operator, ext is not idempotent, but the following holds:

• ext(ext(S)) is a superset of int(S).