Interior (topology)

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The point x is an interior point of S. The point y is on the boundary of S. Interior illustration.svg
The point x is an interior point of S. The point y is on the boundary of S.

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.

Contents

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

Interior point

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

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

a
{\displaystyle a}
is an interior point of
M
{\displaystyle M}
because there is an e-neighbourhood of a which is a subset of
M
.
{\displaystyle M.} Set of real numbers with epsilon-neighbourhood.svg
is an interior point of because there is an ε-neighbourhood of a which is a subset of

On the set of real numbers, one can put other topologies rather than the standard one:

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.

Properties

Let be a topological space and let and be subsets of

Other properties include:

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

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

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

Exterior of a set

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:

Interior-disjoint shapes

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. Interior-disjoint.svg
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

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References

  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