In mathematics, more specifically in point-set topology, the derived set of a subset
S
{\displaystyle S}
of a topological space is the set of all limit points of
S
.
{\displaystyle S.}
It is usually denoted by
S
′
.
{\displaystyle S'.}
The concept was first introduced by Georg Cantor in 1872 and he developed set theory in large part to study derived sets on the real line.
Contents
Definition
The derived set of a subset
S
{\displaystyle S}
of a topological space
X
,
{\displaystyle X,}
denoted by
S
′
,
{\displaystyle S',}
is the set of all points
x
∈
X
{\displaystyle x\in X}
that are limit points of
S
,
{\displaystyle S,}
that is, points
x
{\displaystyle x}
such that every neighbourhood of
x
{\displaystyle x}
contains a point of
S
{\displaystyle S}
other than
Examples
If
R
{\displaystyle \mathbb {R} }
is endowed with its usual Euclidean topology then the derived set of the half-open interval
[
0
,
1
)
{\displaystyle [0,1)}
is the closed interval
[
0
,
1
]
.
{\displaystyle [0,1].}
Consider
R
{\displaystyle \mathbb {R} }
with the topology (open sets) consisting of the empty set and any subset of
R
{\displaystyle \mathbb {R} }
that contains 1. The derived set of
A
:=
{
1
}
Properties
Let
X
{\displaystyle X}
denote a topological space in what follows.
If
A
{\displaystyle A}
and
B
{\displaystyle B}
are subsets of
X
,
{\displaystyle X,}
the derived set has the following properties:
∅
′
=
∅
{\displaystyle \varnothing '=\varnothing }
a
∈
A
′
{\displaystyle a\in A'}
implies
a
∈
(
A
∖
{
a
Closedness of derived sets
The derived set of a set need not be closed in general. For example, if
X
=
{
a
,
b
}
{\displaystyle X=\{a,b\}}
with the indiscrete topology, the set
S
=
{
a
}
{\displaystyle S=\{a\}}
has derived set
S
′
=
{
b
}
,
{\displaystyle S'=\{b\},}
which is not closed in
X
.
{\displaystyle X.}
But the derived set of a closed set is always closed.
For a point
More properties
Two subsets
S
{\displaystyle S}
and
T
{\displaystyle T}
are separated precisely when they are disjoint and each is disjoint from the other's derived set
S
′
∩
T
=
∅
=
T
′
∩
S
.
{\textstyle S'\cap T=\varnothing =T'\cap S.}
A bijection between two topological spaces is a homeomorphism if and only if the derived set of the image (in the second space) of any subset of the first space is the image of the derived set of that subset.
In a T1 space, the derived set of any finite set is empty and furthermore,
(
S
−
{
p
}
)
Topology in terms of derived sets
Because homeomorphisms can be described entirely in terms of derived sets, derived sets have been used as the primitive notion in topology. A set of points
X
{\displaystyle X}
can be equipped with an operator
S
↦
S
∗
{\displaystyle S\mapsto S^{*}}
mapping subsets of
X
{\displaystyle X}
to subsets of
X
,
{\displaystyle X,}
such that for any set
S
{\displaystyle S}
and any point
a
{\displaystyle a}
:
∅
∗
=
∅
{\displaystyle \varnothing ^{*}=\varnothing }
S
Cantor–Bendixson rank
For ordinal numbers
α
,
{\displaystyle \alpha ,}
the
α
{\displaystyle \alpha }
-th Cantor–Bendixson derivative of a topological space X is defined by repeatedly applying the derived set operation using transfinite recursion as follows:
X
0
=
X
{\displaystyle \displaystyle X^{0}=X}
X
α
+
1
=
(
X
α
)
′
{\displaystyle \displaystyle X^{\alpha +1}=\left(X^{\alpha }\right)'}
X
λ
=
⋂
α
<
λ