Interior (topology)
In mathematics, specifically in topology, the interior of a subset of a topological space is the union of all open subsets of that set. A point that is in the interior of S is an interior point of S.
The interior of S is the complement of the closure of the complement of S. In this sense interior and closure are dual notions.
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 are always open while the boundary is always closed. Sets with empty interior have been called boundary sets.^{[1]}
Contents
Definitions
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 completely contained in S. (This is illustrated in the introductory section to this article.)
This definition generalizes to any subset S of a metric space X with metric d: 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 "open set". Let S be a subset of a topological space X. Then x is an interior point of S if x is contained in an open subset of X which is completely contained in S. (Equivalently, x is an interior point of S if S is a neighbourhood of x.)
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.
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.
Examples
- In any space, the interior of the empty set is the empty set.
- In any space X, if , then int(A) is contained in A.
- If X is the Euclidean space of real numbers, then int([0, 1]) = (0, 1).
- If X is the Euclidean space , then the interior of the set of rational numbers is empty.
- If X is the complex plane , then
- 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 , where has the lower limit topology, then int([0, 1]) = [0, 1).
- If one considers on the topology in which every set is open, then int([0, 1]) = [0, 1].
- If one considers on the topology in which the only open sets are the empty set and 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.
Interior operator
The interior operator ^{o} is dual to the closure operator ^{—}, in the sense that
- ,
and also
- ,
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.
Exterior of a set
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).
Interior-disjoint shapes
Two shapes a and b 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
- ^ Kuratowski, Kazimierz (1922). "Sur l'Operation Ā de l'Analysis Situs" (PDF). Fundamenta Mathematicae. Warsaw: Polish Academy of Sciences. 3: 182–199. ISSN 0016-2736.
External links
- Interior at PlanetMath.org.