nLab compact ordered space

Idea

Compact ordered spaces can be thought of as a refinement of the notion of compact Hausdorff space to ordered spaces. They share many properties of compact Hausdorff spaces, where equality is replaced by the order relation. These spaces are a convenient choice for notions of convergence in an ordered setting.

In particular, just as compact Hausdorff spaces can be seen as algebras of the ultrafilter monad on Set, compact ordered spaces can be seen as algebras of the prime upper filter monad? on Pos, (see below).

The concept was introduced by Nachbin (see Nachbin ‘65).

Definition

A compact ordered space or compact pospace is a compact topological space $X$ equipped with a partial order which is closed as a subset of $X\times X$.

Properties

• A compact ordered space is always Hausdorff, i.e. it is a compactum. To see this, note that since the relation is a closed subset of $X\times X$, so is the opposite relation, and hence their intersection too, which is the diagonal of $X\times X$.

• Conversely, a compact Hausdorff space can be seen as a compact ordered space with the discrete order?.

• In a compact ordered space, the up-sets and down-sets $\uparrow\!\{x\}$ and $\downarrow\!\{x\}$ are always closed and compact.

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Categories of compact ordered spaces

A canonical choice of morphisms between compact ordered spaces is continuous, monotone maps, which form a category. This category is usually just called the category of compact ordered spaces, and denoted by $CompOrd$.

With the 2-cell structure given by the pointwise order, $CompOrd$ becomes a locally posetal 2-category.

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As algebras

For now, see Flagg ‘96.

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Connection with stably compact spaces

For now, see Jung ‘04.

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See also

• compact Hausdorff space
• stably compact space

References

• Leopoldo Nachbin, Topology and Order, Van Nostrand, 1965.

• Bob Flagg, Algebraic theories of compact pospaces, Topology and its Applications, 1996 (doi:10.1016/S0166-8641(96)00117-4).

• Achim Jung, Stably compact spaces and the probabilistic powerspace construction, ENTCS 87, 2004 (doi:10.1016/j.entcs.2004.10.001).