Partially ordered space explained
equipped with a closed partial order
, i.e. a partial order whose graph
is a closed subset of
.
From pospaces, one can define dimaps, i.e. continuous maps between pospaces which preserve the order relation.
Equivalences
For a topological space
equipped with a partial order
, the following are equivalent:
is a partially ordered space.
with
, there are open sets
with
and
for all
.
with
, there are disjoint neighbourhoods
of
and
of
such that
is an
upper set and
is a lower set.The
order topology is a special case of this definition, since a
total order is also a partial order.
Properties
Every pospace is a Hausdorff space. If we take equality
as the partial order, this definition becomes the definition of a Hausdorff space.
Since the graph is closed, if
\left(x\alpha\right)\alpha
and
\left(y\alpha\right)\alpha
are
nets converging to
x and
y, respectively, such that
for all
, then
.
External links
Notes and References
- Book: Continuous Lattices and Domains. Gierz. G.. Hofmann. K. H.. Keimel. K.. Lawson. J. D.. Mislove. M.. Scott. D. S.. 2009. 10.1017/CBO9780511542725. 9780521803380 .
