In mathematics, a partially ordered space (or pospace) is a topological space
X
equipped with a closed partial order
≤
, i.e. a partial order whose graph
{
(
x
,
y
)
∈
X
2
|
x
≤
y
}
is a closed subset of
X
2
.
From pospaces, one can define dimaps, i.e. continuous maps between pospaces which preserve the order relation.
For a topological space
X
equipped with a partial order
≤
, the following are equivalent:
X
is a partially ordered space.
For all
x
,
y
∈
X
with
x
≰
y
, there are open sets
U
,
V
⊂
X
with
x
∈
U
,
y
∈
V
and
u
≰
v
for all
u
∈
U
,
v
∈
V
.
For all
x
,
y
∈
X
with
x
≰
y
, there are disjoint neighbourhoods
U
of
x
and
V
of
y
such that
U
is an upper set and
V
is a lower set.
The order topology is a special case of this definition, since a total order is also a partial order. Every pospace is a Hausdorff space. If we take equality
=
as the partial order, this definition becomes the definition of a Hausdorff space.
