In mathematics, the interior of a set S consists of all points which are intuitively "not on the edge of S". A point which is in the interior of S is an interior point of S. The notion of interior is in many ways dual to the notion of closure. Euclid, detail from The School of Athens by Raphael. ... This is a glossary of some terms used in the branch of mathematics known as topology. ... In mathematics, the closure of a set S consists of all points which are intuitively close to S. A point which is in the closure of S is a point of closure of S. The notion of closure is in many ways dual to the notion of interior. ...

Definitions

Interior point

If S is a subset of an Euclidean space, then x is an interior point of S if there exists an open ball centered at x which is contained in S. In mathematics, Euclidean space is a generalization of the 2- and 3-dimensional spaces studied by Euclid. ... A synonym for ball (in geometry or topology, and in any dimension) is disk (or disc Geometry In metric geometry, a ball is a set containing all points within a specified distance of a given point. ...

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. In mathematics, a metric space is a set where a notion of distance between elements of the set is defined. ...

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. Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... This is a glossary of some terms used in the branch of mathematics known as topology. ...

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 S^o. 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. In topology and related fields of mathematics, a set U is called open if, intuitively speaking, you can wiggle or change any point x in U by a small amount in any direction and still be inside U. In other words, if x is surrounded only by elements of U... It has been suggested that this article or section be merged with Logical biconditional. ... In mathematics, an idempotent element is an element which, intuitively, leaves something unchanged. ...

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". For more on this matter, see interior operator below. In mathematics, the interior of a set S consists of all points which are intuitively not on the edge of S. A point which is in the interior of S is an interior point of S. The notion of interior is in many ways dual to the notion of closure. ...

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 R of real numbers, then int([0, 1]) = (0, 1).
• If X is the Euclidean space R, then the interior of the set Q of rational numbers is empty.
• If X is the complex plane C = R², then int({z in C : |z| ≥ 1}) = {z in 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. In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. ... In mathematics, a rational number (or informally fraction) is a ratio or quotient of two integers, usually written as the vulgar fraction a/b, where b is not zero. ... Wikibooks Algebra has more about this subject: Complex numbers In mathematics, a complex number is an expression of the form where a and b are real numbers, and i is a specific imaginary number, called the imaginary unit, with the property i 2 = −1. ... In mathematics, a set is called finite if and only if there is a bijection between the set and some set of the form {1, 2, ..., n} where is a natural number. ...

• If X = R, where R has the lower limit topology, then int([0, 1]) = [0, 1).
• If one considers on R the topology in which every set is open, then int([0, 1]) = [0, 1].
• If one considers on R the topology in which the only open sets are the empty set and 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 mathematics, the lower limit topology or right half-open interval topology is a topology defined on the set R of real numbers; it is different from the standard topology on R and has a number of interesting properties. ...

• 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.

In topology and related fields of mathematics, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points are isolated from each other in a certain sense. ... In topology, a topological space with the trivial topology is one where the only open sets are the empty set and the entire space. ... A is a subset of B, and B is a superset of A. In mathematics, especially in set theory, a set A is a subset of a set B, if A is contained inside B. The relationship of one set being a subset of another is called inclusion. ...

Interior operator

The interior operator ^o is dual to the closure operator ⁻, in the sense that In mathematics, the closure of a set S consists of all points which are intuitively close to S. A point which is in the closure of S is a point of closure of S. The notion of closure is in many ways dual to the notion of interior. ...

S^o = 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. Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... In set theory and other branches of mathematics, two kinds of complements are defined, the relative complement and the absolute complement. ...

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 topology and related branches of mathematics, the Kuratowski closure axioms are a set of axioms which can be used to define a topological structure on a set. ...

See also: interior algebra. In abstract algebra, an interior algebra is an algebraic structure of the signature <A, ·, +, , 0, 1, I> where <A, ·, +, , 0, 1> is a Boolean algebra and I is a unary operator, the interior operator, satisfying the identities: xI ≤ x xII = xI (xy)I = xIyI 1I = 1 xI is called the...