Everything about Closure Topology totally explained
In
mathematics, the
closure of a set
S consists of all
points which are intuitively "close to
S". A point which is in the closure of
S is a
point of closure of
S. The notion of closure is in many ways
dual to the notion of
interior.
Definitions
Point of closure
For
S a subset of an
Euclidean space,
x is a point of closure of
S if every
open ball centered at
x contains a point of
S. (This point may be
x itself and
x needn't be in
S.)
This definition generalises to any subset
S of a
metric space X. Fully expressed, for
X a metric space with metric
d,
x is a point of closure of
S if for every
r > 0, there's a
y in
S such that the distance
d(
x,
y) <
r. (Again, we may have
x =
y.) Another way to express this is to say that
x is a point of closure of
S if the distance
d(
x,
S) :=
inf, then
S is closed in
Q, and the closure of
S in
Q is
S; however, the closure of
S in the Euclidean space
R is the set of all
real numbers greater than
or equal to
Closure operator
The
closure operator − is
dual to the
interior operator
o, in the sense that
» S− =
X (
X S)
o
and also
» So =
X (
X S)
−
where
X denotes 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.
At most 14 distinct sets can be obtained by repeatedly applying the operations of closure and complement to a given set, a result known as the
Kuratowski's closure-complement problem.
Facts about closures
The set
is
closed if and only if
. In particular, the closure of the
empty set is the empty set, and the closure of
itself is
. The closure of an
intersection of sets is always a
subset of (but need not be equal to) the intersection of the closures of the sets. In a
union of
finitely many sets, the closure of the union and the union of the closures are equal; the union of zero sets is the empty set, and so this statement contains the earlier statement about the closure of the empty set as a special case. The closure of the union of infinitely many sets need not equal the union of the closures, but it's always a
superset of the union of the closures.
If
is a
subspace of
containing
, then the closure of
computed in
is equal to the intersection of
and the closure of
computed in
:
. In particular,
is dense in
iff is a subset of
.
Further Information
Get more info on 'Closure Topology'.
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