Interior (topology)

  (Redirected from Interior point)

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 open in X.
  • If T is open in X then TS if and only if T ⊆ Int S.
  • Int S is an open subset of S when S is given the subspace topology.
  • S is an open subset of X if and only if S = int S.
  • Intensive: Int SS.
  • Idempotence: Int(Int S) = Int S.
  • Preserves/distributes over binary intersection: Int (ST) = (Int S) ∩ (Int T).
  • Monotone/nondecreasing with respect to : If ST then Int S ⊆ Int T.

The above statements will remain true if all instances of the symbols/words

"interior", "Int", "open", "subset", and "largest"

are respectively replaced by

"closure", "Cl", "closed", "superset", and "smallest"

and the following symbols are swapped:

  1. "⊆" swapped with "⊇"
  2. "∪" swapped with "∩"

For more details on this matter, see interior operator below or the article Kuratowski closure axioms.

Other properties include:

  • If S is closed in X and Int T = ∅ then Int (ST) = Int S.[2]

Interior operatorEdit

The interior operator   is dual to the closure operator, which is denoted by   or by an overline , in the sense that

 

and also

 

where   is the topological space containing   and the backslash   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[3] (C. Ursescu) — Let   be a sequence of subsets of a complete metric space  

  • If each   is closed in   then
     
  • If each   is open in   then
     

Exterior of a setEdit

The (topological) exterior of a subset   of a topological space   denoted by   or simply   is the complement of the closure of  :

 

although it can be equivalently defined in terms of the interior by:

 

Alternatively, the interior   could instead be defined in terms of the exterior by using the set equality

 

As a consequence of this relationship between the interior and exterior, many properties of the exterior   can be readily deduced directly from those of the interior   and elementary set identities. Such properties include the following:

  •   is an open subset of   that is disjoint from  
  • If   then  
  •   is equal to the union of all open subsets of   that are disjoint from  
  •   is equal to the largest open subset of   that is disjoint from  

Unlike the interior operator,   is not idempotent, although it does have the property that  

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 & Beckenstein 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.

BibliographyEdit

External linksEdit