Interior (topology)
In
mathematics, the
interior of a set
S consists of all
points which are intuitively "not on the edge of
S". A point which is in the interior of
S is an
interior point of
S. The notion of interior is in many ways dual to the notion of
closure.
Interior point
If
S is a subset of an
Euclidean space, then
x is an interior point of
S if there exists an
open ball centered at
x which is contained in
S.
This definition generalises to any subset
S of a
metric space X. Fully expressed, if
X is a metric space with metric
d, then
x is an interior point of
S if there exists
r > 0, such that
y is in
S whenever the distance
d(
x,
y) <
r.
This definition generalises to
topological spaces by replacing "open ball" with "
neighbourhood". Let
S be a subset of a topological space
X. Then
x is an interior point of
S if there exists a neighbourhood of
x which is contained in
S. Note that this definition does not depend upon whether neighbourhoods are required to be open.
Interior of a set
The
interior of a set
S is the set of all interior points of
S. The interior of
S is denoted int(
S), Int(
S), or
So. The interior of a set has the following properties.
*int(
S) is an open subset of
S.
*int(
S) is the union of all
open sets contained in
S.
*int(
S) is the largest open set contained in
S.
*A set
S is open
if and only if S = int(
S).
*int(int(
S)) = int(
S). (
idempotence)
*If
S is a subset of
T, then int(
S) is a subset of int(
T).
*If
A is an open set, then
A is a subset of
S if and only if
A is a subset of int(
S).
Sometimes the second or third property above is taken as the
definition of the topological interior.
Note that these properties are also satisfied if "interior", "subset", "union", "contained in", "largest" and "open" are replaced by "closure", "superset", "intersection", "which contains", "smallest", and "closed". For more on this matter, see
interior operator below.
*In any space, the interior of the empty set is the empty set.
*In any space
X, int(
X) is contained in
X.
*If
X is the Euclidean space
R of
real numbers, then int([0, 1]) = (0, 1).
*If
X is the Euclidean space
R, then the interior of the set
Q of
rational numbers is empty.
*If
X is the
complex plane C =
R2, then int({
z in
C : |
z| ≥ 1}) = {
z in
C : |
z| > 1}.
*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
X =
R, where
R has the
lower limit topology, then int([0, 1]) =
[0, 1).
*If one considers on
R the topology in which every set is open, then int([0, 1]) = [0, 1].
*If one considers on
R the topology in which the only open sets are the empty set and
R itself, then int([0, 1]) 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 X, since the only open sets are the empty set and
X itself, we have int(
X) =
X and for every
proper subset A of
X, int(
A) is the empty set.
The
interior operator o is dual to the
closure operator
−, in the sense that
So =
X \ (
X \
S)
−,
and also
S− =
X \ (
X \
S)
owhere
X is the
topological space containing
S, and the backslash refers to the
set-theoretic difference.
Therefore, the abstract theory of closure operators and the
Kuratowski closure axioms can be easily translated into the language of interior operators, by replacing sets with their complements.
See also:
interior algebra.
