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 of Priestley spaces and the category of bounded distributive lattices. 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 , 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.) 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 or . Assuming, without loss of generality, that , (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. To see this, one proceeds as follows. For each y ∈ X − U, either or . By the Priestley separation axiom, there exists a clopen up-set or a clopen down-set containing x and missing y. The intersection of these clopen neighborhoods of x does not meet X − U. Therefore, as X is compact, there exists a finite intersection of these clopen neighborhoods of x missing X − U. This finite intersection is the desired clopen neighborhood C of x contained in U. It follows that for each Priestley space (X,τ,≤), the topological space (X,τ) is a Stone space; that is, it is a compact Hausdorff zero-dimensional space. Some further useful properties of Priestley spaces are listed below. Let (X,τ,≤) be a Priestley space. (a) For each closed subset F of X, both ↑ F = {x ∈ X : y ≤ x for some y ∈ F} and ↓ F = { x ∈ X : x ≤ y for some y ∈ F} are closed subsets of X.