ordered space

Definition. A set X that is both a topological spaceMathworldPlanetmath 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 X. In other words, the topology 𝒯 and the partial ordering on X operate independently of one another.

If the partial order is a total orderMathworldPlanetmath, then X 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 structuresMathworldPlanetmath.

  1. 1.

    For example, any topological space is trivially an ordered space, with the partial order defined by ab iff a=b. But this is not so interesting. A more interesting example is to take a T0 space X, and define ab iff a{b}¯. The relationMathworldPlanetmath so defined turns out to be a partial order on X, called the specialization order, making X an ordered space.

  2. 2.

    On the other hand, given any poset P, 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.

  3. 3.

    A slightly more useful example is to take a poset P, and take


    the family of all set complementsPlanetmathPlanetmath of principal upper sets of P, as the subbasis for the topology ω(P) of P. The topology ω(P) so generated is called the lower topology on P.

  4. 4.

    Dually, if we take


    as the subbasis, we get the upper topology on P, denoted by ν(P).

  5. 5.

    In the lower topology ω(P) of P, if yP-x, then either y<x (strict inequalityMathworldPlanetmath) or xy (incomparable with x). If x is an isolated element, then P-x=P-{x}. This means that {x} is a closed setPlanetmathPlanetmath. Similarly, {x} is closed in the upper topology ν(P).

    If x is the top element of P, then {x} is a closed set in ω(P), since P-x=P-{x} is open. Similarly {x} is closed in ν(P) if x is the bottom element in P.

    If P is totally ordered, there are no isolated elements. As a result, we may write P-x in a more familiar fashion: (-,x). Similarly, P-x may be written as (x,).

  6. 6.

    Things get more interesting when we take the common refinement of ω(P) and ν(P). What we end up with is called the interval topology of P.

    When P is totally ordered, the interval topology on P has


    as a subbasis, where (x,y) denotes the open poset interval, consisting of elements aP such that x<a<y. Since finite intersectionsMathworldPlanetmath of open poset intervals is a poset interval, an open set in P 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 X be an ordered space. Below are some of the common conditions that can be imposed on X:

  • X is said to be upper semiclosed if x is a closed set for every xX.

  • Similarly, X is lower semiclosed if x is closed in X.

  • X is semiclosed if it is both upper and lower semiclosed.

  • If , as a subset of X×X, is closed in the product topology, then X 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 structuresPlanetmathPlanetmath 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, ContinuousMathworldPlanetmathPlanetmath Lattices and Domains, Cambridge University Press, Cambridge (2003).
Title ordered space
Canonical name OrderedSpace
Date of creation 2013-03-22 17:05:36
Last modified on 2013-03-22 17:05:36
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 15
Author CWoo (3771)
Entry type Definition
Classification msc 54E99
Classification msc 06F20
Classification msc 06F30
Synonym ordered topological space
Synonym topological ordered space
Synonym partially ordered space
Synonym partially ordered topological space
Related topic OrderTopology
Defines upper topology
Defines lower topology
Defines interval topology
Defines upper semiclosed
Defines lower semiclosed
Defines semiclosed
Defines pospace
Defines totally ordered space