Interior (topology)

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In mathematics, the interior of a set S consists of all points of S that are intuitively "not on the edge of S". A point that is in the interior of S is an interior point of S.

The exterior of a set is the interior of its complement; it consists of the points that are not in the set or its boundary.

The notion of the interior of a set is a topological concept; it is not defined for all sets, but it is defined for sets that are a subset of a topological space. It is in many ways dual to the notion of closure.

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[edit] Definitions

[edit] Interior point

If S is a subset of a Euclidean space, then x is an interior point of S if there exists an open ball centered at x which is contained in S.

This definition generalises to any subset S of a metric space X. Fully expressed, if X is a metric space with metric d, then x is an interior point of S if there exists r > 0, such that y is in S whenever the distance d(x, y) < r.

This definition generalises to topological spaces by replacing "open ball" with "neighbourhood". Let S be a subset of a topological space X. Then x is an interior point of S if there exists a neighbourhood of x which is contained in S. Note that this definition does not depend upon whether neighbourhoods are required to be open. If neighbourhoods are not required to be open then S will automatically be a neighbourhood of x if S contains a neighbourhood of x.

[edit] Interior of a set

The interior of a set S is the set of all interior points of S. The interior of S is denoted int(S), Int(S), or So. The interior of a set has the following properties.

  • int(S) is an open subset of S.
  • int(S) is the union of all open sets contained in S.
  • int(S) is the largest open set contained in S.
  • A set S is open if and only if S = int(S).
  • int(int(S)) = int(S) (idempotence).
  • If S is a subset of T, then int(S) is a subset of int(T).
  • If A is an open set, then A is a subset of S if and only if A is a subset of int(S).

Sometimes the second or third property above is taken as the definition of the topological interior.

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.

[edit] Examples

  • In any space, the interior of the empty set is the empty set.
  • In any space X, int(X) is contained in X.
  • If X is the Euclidean space \mathbb{R} of real numbers, then int([0, 1]) = (0, 1).
  • If X is the Euclidean space \mathbb{R}, then the interior of the set \mathbb{Q} of rational numbers is empty.
  • If X is the complex plane \mathbb{C} = \mathbb{R}^2, then int(\{z\in \mathbb{C} : |z| \geq 1\}) = \{z\in \mathbb{C} : |z| > 1\}.
  • 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.

  • If X = \mathbb{R}, where \mathbb{R} has the lower limit topology, then int([0, 1]) = [0, 1).
  • If one considers on \mathbb{R} the topology in which every set is open, then int([0, 1]) = [0, 1].
  • If one considers on \mathbb{R} the topology in which the only open sets are the empty set and \mathbb{R} itself, then int([0, 1]) 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 X, since the only open sets are the empty set and X itself, we have int(X) = X and for every proper subset A of X, int(A) is the empty set.

[edit] Interior operator

The interior operator o is dual to the closure operator —, in the sense that

So = X \ (X \ S)—,

and also

S— = X \ (X \ S)o

where X is the topological space containing S, 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.

[edit] Exterior of a set

Main article: Exterior (topology)

The exterior of a subset S of a topological space X, denoted ext(S) or Ext(S), is the interior int(X\S) of its relative complement. Alternatively, it can be defined as X\S—, the complement of the closure of S. 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).

[edit] See also

[edit] References