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. A point that is in the interior of S is an interior point of S.

The point x is an interior point of S. The point y is on the boundary 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]

DefinitionsEdit

Interior pointEdit

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 setEdit

The interior of a subset S of a topological space X, denoted by Int S or S°, can be defined in any of the following equivalent ways:

  1. Int S is the largest open subset of X contained (as a subset) in S;
  2. Int S is the union of all open sets of X contained in S;
  3. Int S is the set of all interior points of S.

ExamplesEdit

 
a is an interior point of M, because there is an ε-neighbourhood of a which is a subset of M.
  • In any space, the interior of the empty set is the empty set.
  • In any space X, if SX, then int SS.
  • 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 X = ℝ, 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 X = int X and for every proper subset S of X, int S is the empty set.

PropertiesEdit

Let X be a topological space and let S and T be subset of X.

  • Int S is an open subset of S and an open subset of X.
  • S is an open subset of X if and only if S = int S.
  • Int(Int S) = Int S (idempotence).
  • If ST then Int S ⊆ Int T.
  • If A is an open subset of X, then AS if and only if A ⊆ Int S.
  • If S is closed in X and Int T = ∅ then Int (ST) = Int S.[2]

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.

Interior operatorEdit

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.

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

Theorem[3] (C. Ursescu) — Let X be a complete metric space and let   be a sequence of subsets of X.

  • If each Si is closed in X then  .
  • If each Si is open in X then  .

Exterior of a setEdit

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 ST, 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 shapesEdit

 
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.

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 alsoEdit

ReferencesEdit

  1. ^ 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.
  2. ^ Narici 2011, pp. 371-423.
  3. ^ Zalinescu, C (2002). Convex analysis in general vector spaces. River Edge, N.J. London: World Scientific. p. 33. ISBN 981-238-067-1. OCLC 285163112.

External linksEdit