Definition. A set that is both a topological space and a poset is variously called a topological ordered space, ordered topological space, or simply an ordered space. Note that there is no compatibility conditions imposed on . In other words, the topology and the partial ordering on operate independently of one another.
If the partial order is a total order, then is called a totally ordered space. In some literature, a totally ordered space is called an ordered space. In this entry, however, an ordered space is always a partially ordered space.
One can construct an ordered space from a set with fewer structures.
For example, any topological space is trivially an ordered space, with the partial order defined by iff . But this is not so interesting. A more interesting example is to take a space , and define iff . The relation so defined turns out to be a partial order on , called the specialization order, making an ordered space.
On the other hand, given any poset , we can arbitrarily assign a topology on it, making it an ordered space, so that every poset is trivially an ordered space. Again this is not very interesting.
Dually, if we take
as the subbasis, we get the upper topology on , denoted by .
In the lower topology of , if , then either (strict inequality) or (incomparable with ). If is an isolated element, then . This means that is a closed set. Similarly, is closed in the upper topology .
If is the top element of , then is a closed set in , since is open. Similarly is closed in if is the bottom element in .
If is totally ordered, there are no isolated elements. As a result, we may write in a more familiar fashion: . Similarly, may be written as .
Things get more interesting when we take the common refinement of and . What we end up with is called the interval topology of .
When is totally ordered, the interval topology on has
as a subbasis, where denotes the open poset interval, consisting of elements such that . Since finite intersections of open poset intervals is a poset interval, an open set in can be written as an (arbitrary) union of open poset intervals.
As an example, the usual topology on is precisely the interval topology generated by the linear order on .
Remark. It is a common practice in mathematics to impose special compatibility conditions on a structure having two inherent substructures so the substructures inter-relate, so that one can derive more interesting fruitful results. This is true also in the case of an ordered space. Let be an ordered space. Below are some of the common conditions that can be imposed on :
is said to be upper semiclosed if is a closed set for every .
Similarly, is lower semiclosed if is closed in .
is semiclosed if it is both upper and lower semiclosed.
If , as a subset of , is closed in the product topology, then is called a pospace.
Other structures, such as ordered topological vector spaces, topological lattices (http://planetmath.org/TopologicalLattice), and topological vector lattices (http://planetmath.org/TopologicalVectorLattice) are ordered spaces with algebraic structures satisfying certain additional compatibility conditions. Please click on the links for details.
- 1 G. Gierz, K. H. Hofmann, K. Keimel, J. D. Lawson, M. W. Mislove, D. S. Scott, Continuous Lattices and Domains, Cambridge University Press, Cambridge (2003).
|Date of creation||2013-03-22 17:05:36|
|Last modified on||2013-03-22 17:05:36|
|Last modified by||CWoo (3771)|
|Synonym||ordered topological space|
|Synonym||topological ordered space|
|Synonym||partially ordered space|
|Synonym||partially ordered topological space|
|Defines||totally ordered space|