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.
If
S
x
S
x
S.
This definition generalizes to any subset
S
X
d
x
S
r>0,
y
S
d(x,y)<r.
This definition generalizes to topological spaces by replacing "open ball" with "open set".If
S
X
x
S
X
x
X
S.
x
S
S
x.
The interior of a subset
S
X,
\operatorname{int}XS
\operatorname{int}S
S\circ,
\operatorname{int}S
X
S.
\operatorname{int}S
X
S.
\operatorname{int}S
S.
X
\operatorname{int}S
\operatorname{int}XS.
X,
S\subseteqX,
\operatorname{int}S\subseteqS.
X
\Reals
\operatorname{int}([0,1])=(0,1)
\Q
\operatorname{int}\Q=\varnothing
X
\Complex,
\operatorname{int}(\{z\in\Complex:|z|\leq1\})=\{z\in\Complex:|z|<1\}.
On the set of real numbers, one can put other topologies rather than the standard one:
X
\Reals
\operatorname{int}([0,1])=[0,1).
\Reals
\operatorname{int}([0,1])=[0,1].
\Reals
\Reals
\operatorname{int}([0,1])
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.
X,
X
\operatorname{int}X=X
S
X,
\operatorname{int}S
Let
X
S
T
X.
\operatorname{int}S
X.
T
X
T\subseteqS
T\subseteq\operatorname{int}S.
\operatorname{int}S
S
S
S
X
\operatorname{int}S=S.
\operatorname{int}S\subseteqS.
\operatorname{int}(\operatorname{int}S)=\operatorname{int}S.
\operatorname{int}(S\capT)=(\operatorname{int}S)\cap(\operatorname{int}T).
\operatorname{int}(S\cupT)~\supseteq~(\operatorname{int}S)\cup(\operatorname{int}T)
X=\Reals,S=(-infty,0],
T=(0,infty)
(\operatorname{int}S)\cup(\operatorname{int}T)=(-infty,0)\cup(0,infty)=\Reals\setminus\{0\}
\operatorname{int}(S\cupT)=\operatorname{int}\Reals=\Reals.
S\subseteqT
\operatorname{int}S\subseteq\operatorname{int}T.
Other properties include:
S
X
\operatorname{int}T=\varnothing
\operatorname{int}(S\cupT)=\operatorname{int}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", "cl", "closed", "superset", and "smallest"and the following symbols are swapped:
\subseteq
\supseteq
\cup
\cap
The interior operator
\operatorname{int}X
\operatorname{cl}X
X
S,
\setminus
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.
The exterior of a subset
S
X,
\operatorname{ext}XS
\operatorname{ext}S,
S,
X
S.
Similarly, the interior is the exterior of the complement:
The interior, boundary, and exterior of a set
S
\partialS
S.
Some of the properties of the exterior operator are unlike those of the interior operator:
S\subseteqT,
\operatorname{ext}T\subseteq\operatorname{ext}S.
\operatorname{int}S\subseteq\operatorname{ext}\left(\operatorname{ext}S\right).
Two shapes
a
b