Interior (topology)

Interior (topology)
In mathematics, specifically in topology,
the interior of a subset of a topological space is the union of all subsets of that are open in .
A point that is in the interior of is an interior point of .
The interior of is the complement of the closure of the complement of .
In this sense interior and closure are dual notions.
The exterior of a set is the complement of the closure of ; 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 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:
is the largest open subset of contained in
is the union of all open sets of contained in
is the set of all interior points of
If the space is understood from context then the shorter notation is usually preferred to
Examples
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
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
:
:
/:
However, the interior operator does not distribute over unions since only is guaranteed in general and equality might not hold. For example, if and then is a proper subset of
/: 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:
"" swapped with ""
"" 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:
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; 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 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
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
References
Bibliography
External links
Closure operators
General topology