Interior (topology)

Summary

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 of a closed curve are a slightly different concept; see the Jordan curve theorem.

Definitions edit

Interior point edit

If   is a subset of a Euclidean space, then   is an interior point of   if there exists an open ball centered at   which is completely contained in   (This is illustrated in the introductory section to this article.)

This definition generalizes to any subset   of a metric space   with metric  :   is an interior point of   if there exists a real number   such that   is in   whenever the distance  

This definition generalizes to topological spaces by replacing "open ball" with "open set". If   is a subset of a topological space   then   is an interior point of   in   if   is contained in an open subset of   that is completely contained in   (Equivalently,   is an interior point of   if   is a neighbourhood of  )

Interior of a set edit

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

  1.   is the largest open subset of   contained in  
  2.   is the union of all open sets of   contained in  
  3.   is the set of all interior points of  

If the space   is understood from context then the shorter notation   is usually preferred to  

Examples edit

 
  is an interior point of   because there is an ε-neighbourhood of a which is a subset of  
  • In any space, the interior of the empty set is the empty set.
  • In any space   if   then  
  • If   is the real line   (with the standard topology), then   whereas the interior of the set   of rational numbers is empty:  
  • If   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   is the real numbers   with the lower limit topology, then  
  • If one considers on   the topology in which every set is open, then  
  • If one considers on   the topology in which the only open sets are the empty set and   itself, then   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   since the only open sets are the empty set and   itself,   and for every proper subset   of     is the empty set.

Properties edit

Let   be a topological space and let   and   be subsets of  

  •   is open in  
  • If   is open in   then   if and only if  
  •   is an open subset of   when   is given the subspace topology.
  •   is an open subset of   if and only if  
  • Intensive:  
  • Idempotence:  
  • Preserves/distributes over binary intersection:  
    • However, the interior operator does not distribute over unions since only   is guaranteed in general and equality might not hold.[note 1] For example, if   and   then   is a proper subset of  
  • Monotone/nondecreasing with respect to  : If   then  

Other properties include:

  • If   is closed in   and   then  

Relationship with closure

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.

Interior operator edit

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[1] (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
     

The result above implies that every complete metric space is a Baire space.

Exterior of a set edit

The exterior of a subset   of a topological space   denoted by   or simply   is the largest open set disjoint from   namely, it is the union of all open sets in   that are disjoint from   The exterior is the interior of the complement, which is the same as the complement of the closure;[2] in formulas,

 

Similarly, the interior is the exterior of the complement:

 

The interior, boundary, and exterior of a set   together partition the whole space into three blocks (or fewer when one or more of these is empty):

 
where   denotes the boundary of  [3] The interior and exterior are always open, while the boundary is closed.

Some of the properties of the exterior operator are unlike those of the interior operator:

  • The exterior operator reverses inclusions; if   then  
  • The exterior operator is not idempotent. It does have the property that  

Interior-disjoint shapes edit

 
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   and   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 edit

References edit

  1. ^ 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.
  2. ^ Bourbaki 1989, p. 24.
  3. ^ Bourbaki 1989, p. 25.
  1. ^ The analogous identity for the closure operator is   These identities may be remembered with the following mnemonic. Just as the intersection   of two open sets is open, so too does the interior operator distribute over intersections   explicitly:   And similarly, just as the union   of two closed sets is closed, so too does the closure operator distribute over unions   explicitly:  

Bibliography edit

External links edit