Interior (topology)

In mathematics, specifically in topology, the interior of a subset S of a topological space X is the union of all subsets of S that are open in X.

The exterior of a set S is the complement of the closure of S; 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).

The interior and exterior of a closed curve are a slightly different concept; see the Jordan curve theorem.

is a subset of a Euclidean space, then

if there exists an open ball centered at

(This is illustrated in the introductory section to this article.)

This definition generalizes to any subset

This definition generalizes to topological spaces by replacing "open ball" with "open set".

is a subset of a topological space

is contained in an open subset of

can be defined in any of the following equivalent ways: If the space

is understood from context then the shorter notation

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

Other properties include: Relationship with closure The above statements will remain true if all instances of the symbols/words are respectively replaced by and the following symbols are swapped: For more details on this matter, see interior operator below or the article Kuratowski closure axioms.

is dual to the closure operator, which is denoted by

denotes set-theoretic difference.

Therefore, the abstract theory of closure operators and the Kuratowski closure axioms can be readily translated into the language of interior operators, by replacing sets with their complements in

In general, the interior operator does not commute with unions.

However, in a complete metric space the following result does hold: Theorem[1] (C. Ursescu) — Let

be a sequence of subsets of a complete metric space

is the largest open set disjoint from

namely, it is the union of all open sets in

Similarly, the interior is the exterior of the complement:

The interior, boundary, and exterior of a set

together partition the whole space into three blocks (or fewer when one or more of these is empty):

[3] The interior and exterior are always open, while the boundary is closed.

are called interior-disjoint if the intersection of their interiors is empty.

Interior-disjoint shapes may or may not intersect in their boundary.

The point x is an interior point of S . The point y is on the boundary of S .
is an interior point of because there is an ε-neighbourhood of a which is a subset of
The red shapes are not interior-disjoint with the blue Triangle. The green and the yellow shapes are interior-disjoint with the blue Triangle, but only the yellow shape is entirely disjoint from the blue Triangle.