nLab
compact ordered space

Contents

Context

Topology

topology (point-set topology, point-free topology)

see also differential topology, algebraic topology, functional analysis and topological homotopy theory

Introduction

Basic concepts

Universal constructions

Extra stuff, structure, properties

Examples

Basic statements

Theorems

Analysis Theorems

topological homotopy theory

(0,1)(0,1)-Category theory

Contents

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 XX equipped with a partial order which is closed as a subset of X×XX\times X.

Properties

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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 CompOrdCompOrd.

With the 2-cell structure given by the pointwise order, CompOrdCompOrd 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

References

Last revised on October 25, 2019 at 11:25:35. See the history of this page for a list of all contributions to it.