Priestley space

In mathematics, a Priestley space is an ordered topological space with special properties. Priestley spaces are named after Hilary Priestley who introduced and investigated them. Priestley spaces play a fundamental role in the study of distributive lattices. In particular, there is a duality ("Priestley duality") between the category of Priestley spaces and the category of bounded distributive lattices.

Definition

A Priestley space is an ordered topological space (X,τ,≤), i.e. a set X equipped with a partial order ≤ and a topology τ, satisfying

the following two conditions:

(X,τ) is compact.

If

x

≰

y

{\displaystyle \scriptstyle x\,\not \leq \,y}

, then there exists a clopen up-set U of X such that x∈U and y∉ U. (This condition is known as the Priestley separation axiom.)

Properties of Priestley spaces

Each Priestley space is Hausdorff. Indeed, given two points x,y of a Priestley space (X,τ,≤), if x≠ y, then as ≤ is a partial order, either

x

≰

y

{\displaystyle \scriptstyle x\,\not \leq \,y}

or

y

≰

x

{\displaystyle \scriptstyle y\,\not \leq \,x}

. Assuming, without loss of generality, that

x

≰

y

{\displaystyle \scriptstyle x\,\not \leq \,y}

, (ii) provides a clopen up-set U of X such that x∈ U and y∉ U. Therefore, U and V = X − U are disjoint open subsets of X separating x and y.

Each Priestley space is also zero-dimensional; that is, each open neighborhood U of a point x of a Priestley space (X,τ,≤) contains a clopen neighborhood C of x.