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 (or fewer when one or more of these is empty).

Definitions

Interior point

is a subset of a Euclidean space if there exists an open ball (This is illustrated in the introductory section to this article.)

of a metric space

This definition generalizes to topological spaces by replacing "open ball" with "open set is an is a neighbourhood)

Interior of a set

The interior can be defined in any of the following equivalent ways:

Examples

• In any space, the interior of the empty set is the empty set.

• is the real line of rational numbers

• is the complex plane

• In any Euclidean space, the interior of any finite set is the empty set.

On the set of real numbers, one can put other topologies rather than the standard one:

• with the lower limit topology

• the topology in which every set is open

• 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.

• In any discrete space, since every set is open, every set is equal to its interior.

• In any indiscrete space and for every proper subset is the empty set.

Properties

• is open

• is given the subspace topology.

• if and only if

• Idempotence

/

• is guaranteed in general and equality might not hold.

• /

Other properties include:

Relationship with closure

The above statements will remain true if all instances of the symbols/words

1. "interior", "int", "open", "subset", and "largest"

are respectively replaced by

1. "closure", "cl", "closed", "superset", and "smallest"

and the following symbols are swapped:

1. "

2. "

For more details on this matter, see interior operator below or the article Kuratowski closure axioms.

Interior operator

The interior operator is dual to the closure operator or by an overline — is the topological space denotes set-theoretic difference. Therefore, the abstract theory of closure operators and the Kuratowski closure axioms

In general, the interior operator does not commute with unions. However, in a complete metric space the following result does hold:

The result above implies that every complete metric space is a Baire space.

Exterior of a set

The exterior is the largest open set disjoint The exterior is the interior of the complement, which is the same as the complement of the closure;

The interior, boundary together partition The interior and exterior are always open, while the boundary is closed.

Some of the properties of the exterior operator are unlike those of the interior operator:

• The exterior operator is not idempotent

Interior-disjoint shapes

are called interior-disjoint if the intersection of their interiors is empty. Interior-disjoint shapes may or may not intersect in their boundary.

See also

References

Bibliography

External links