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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).

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 $r > 0,$ such that is in whenever the distance $d(x, y) < r.$

This definition generalises to topological spaces by replacing "open ball" with " open set".
Let be a subset of a topological space .
Then is an interior point of if is contained in an open subset of which 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 $\operatornameS$ or $\operatornameS$ or $S^\circ,$ can be defined in any of the following equivalent ways:
# is the largest open subset of contained (as a subset) in
# is the union of all open sets of contained in
# is the set of all interior points of

Examples

*In any space, the interior of the empty set is the empty set.
*In any space , if $S \subseteq X,$ then $\operatorname S \subseteq S.$
*If is the real line $\Reals$ (with the standard topology), then .
*If is the real line $\Reals,$ then the interior of the set $\Q$ of rational numbers is empty.
*If is the complex plane $\Complex,$ then $\operatorname(\) = \.$
*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 $\Reals$ with the lower limit topology, then .
*If one considers on $\Reals$ the topology in which every set is open, then .
*If one considers on $\Reals$ the topology in which the only open sets are the empty set and $\Reals$ 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, $\operatorname X = X$ and for every proper subset of , $\operatorname S$ is the empty set.

Properties

Let be a topological space and let and be subsets of .

* $\operatorname S$ is open in .
* If is open in then $T \subseteq S$ if and only if $T \subseteq \operatorname S.$
* $\operatorname S$ is an open subset of when is given the subspace topology.
* is an open subset of if and only if $\operatorname S = S.$
* : $\operatorname S \subseteq S.$
* : $\operatorname (\operatorname S) = \operatorname S.$
* /: $\operatorname (S \cap T) = (\operatorname S) \cap (\operatorname T).$
** However, the interior operator does not distribute over unions since only $\operatorname (S \cup T) ~\supseteq~ (\operatorname S) \cup (\operatorname T)$ is guaranteed in general and equality might not hold. For example, if $X = \Reals, S = (-\infty, 0],$ and $T = (0, \infty)$ then $(\operatorname S) \cup (\operatorname T) = (-\infty, 0) \cup (0, \infty) = \Reals \setminus \$ is a proper subset of $\operatorname (S \cup T) = \operatorname \Reals = \Reals.$
* /: If $S \subseteq T$ then $\operatorname S \subseteq \operatorname T.$

Other properties include:

* If is closed in and $\operatorname T = \varnothing$ then $\operatorname (S \cup T) = \operatorname S.$

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 (topology), closure", "cl", "closed", "superset", and "smallest"
and the following symbols are swapped:
# "$\subseteq$" swapped with "$\supseteq$"
# "$\cup$" swapped with "$\cap$"
For more details on this matter, see interior operator below or the article Kuratowski closure axioms.

Interior operator

The interior operator $\operatorname_X$ is dual to the Closure (topology), closure operator, which is denoted by $\operatorname_X$ or by an overline ^—, in the sense that
$\operatorname_X S = X \setminus \overline$
and also
$\overline = X \setminus \operatorname_X (X \setminus S),$
where $X$ is the topological space containing $S,$ and the backslash $\,\setminus\,$ 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 $X.$

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 $S$ of a topological space $X,$ denoted by $\operatorname_X S$ or simply $\operatorname S,$ is the largest open set disjoint from $S,$ namely, it is the union of all open sets in $X$ that are disjoint from $S.$ The exterior is the interior of the complement, which is the same as the complement of the closure; in formulas,
$\operatornameS = \operatorname(X\setminus S) = X\setminus\overline.$

Similarly, the interior is the exterior of the complement:
$\operatornameS = \operatorname(X \setminus S).$

The interior, boundary, and exterior of a set $S$ together partition the whole space into three blocks (or fewer when one or more of these is empty):
$X = \operatornameS \cup \partial S \cup \operatornameS,$
where $\partial S$ denotes the boundary of $S.$ 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 $S \subseteq T,$ then $\operatornameT \subseteq \operatornameS.$
* The exterior operator is not idempotent. It does have the property that $\operatornameS \subseteq \operatorname\left(\operatornameS\right).$

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

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References

Bibliography

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External links

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{{Topology, expanded

Closure operators
General topology