In mathematics , a partition topology is a topology that can be induced on any set
X
{\displaystyle X}
partitioning
X
{\displaystyle X}
P
;
{\displaystyle P;}
basis for the topology. There are two important examples which have their own names:
The odd–even topology
X
=
N
{\displaystyle X=\mathbb {N} }
P
=
{
{
2
k
−
1
,
2
k
}
:
k
∈
N
}
.
{\displaystyle P={\left\{~\{2k-1,2k\}:k\in \mathbb {N} \right\}}.}
P
=
{
{
1
,
2
}
,
{
3
,
4
}
,
{
5
,
6
}
,
…
}
.
{\displaystyle P=\{~\{1,2\},\{3,4\},\{5,6\},\ldots \}.}
The deleted integer topology
X
=
⋃
n
∈
N
(
n
−
1
,
n
)
⊆
R
{\displaystyle X={\begin{matrix}\bigcup _{n\in \mathbb {N} }(n-1,n)\subseteq \mathbb {R} \end{matrix}}}
P
=
{
(
0
,
1
)
,
(
1
,
2
)
,
(
2
,
3
)
,
…
}
.
{\displaystyle P={\left\{(0,1),(1,2),(2,3),\ldots \right\}}.}
The trivial partitions yield the discrete topology (each point of
X
{\displaystyle X}
P
,
{\displaystyle P,}
P
=
{
{
x
}
:
x
∈
X
}
{\displaystyle P=\{~\{x\}~:~x\in X~\}}
indiscrete topology (the entire set
X
{\displaystyle X}
P
,
{\displaystyle P,}
P
=
{
X
}
{\displaystyle P=\{X\}}
Any set
X
{\displaystyle X}
P
{\displaystyle P}
pseudometric space with a pseudometric given by:
d
(
x
,
y
)
=
{
0
if
x
and
y
are in the same partition element
1
otherwise
.
{\displaystyle d(x,y)={\begin{cases}0&{\text{if }}x{\text{ and }}y{\text{ are in the same partition element}}\\1&{\text{otherwise}}.\end{cases}}}
This is not a metric unless
P
{\displaystyle P}
The partition topology provides an important example of the independence of various separation axioms . Unless
P
{\displaystyle P}
P
{\displaystyle P}
topologically indistinguishable : the topology does not separate points. Hence
X
{\displaystyle X}
Kolmogorov space , nor a T1 space , a Hausdorff space or an Urysohn space . In a partition topology the complement of every open set is also open, and therefore a set is open if and only if it is closed. Therefore,
X
{\displaystyle X}
regular , completely regular , normal and completely normal .
X
/
P
{\displaystyle X/P}
See also
References
