Order, lattices, ordered algebraic structures Definition

Priestley space

Builds on Topological space Compact space Partial order Upper set

Definition

Let $X$ be a set with a topology $\tau$ and a partial order $\le$ . The triple $(X, \tau, \le)$ is a Priestley space if:

$(X, \tau)$ is compact;
for all $x, y \in X$ with $x \not\le y$ , some $U \subseteq X$ has $x \in U$ and $y \notin U$ , is open and closed in $\tau$ , and is an upper set.

Clause 2 is called the Priestley separation axiom.

Remarks

A Priestley space is Hausdorff and totally disconnected, so its underlying space is a Stone space. Indeed, for $x \ne y$ , say $x \not\le y$ , the sets $U$ and $X \setminus U$ from clause 2 separate $x$ and $y$ .

Example: a finite poset with the discrete topology. Every subset is clopen, and for $x \not\le y$ the upper set ${\uparrow} x = \set{z : x \le z}$ contains $x$ but not $y$ .

Non-example: the interval $[0, 1]$ with its usual topology and usual order. It is compact, but its only clopen sets are $\emptyset$ and $[0, 1]$ , so no clopen set separates $1 \not\le 0$ , and clause 2 fails.

The morphisms of Priestley spaces are the continuous order-preserving maps.

Reading path

To understand this page from the ground up, read these 5 pages in order. The foundations come first.

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Compact space concept
Partial order concept
Topological space concept
Upper set concept
Priestley space definition

Tag	`DYM1EFAC59`
URI	`m2://DYM1EFAC59`
Kind	definition
Writing	W0 stub
MSC	06D50, 54F05, 06E15
Defines	Priestley space
Uses	Topological space, Compact space, Partial order, Upper set
Changed	2 November 2026

Priestley space

Definition

Remarks

Reading path

Further reading

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