down-sets of .

::(c) Each closed up-set of is an intersection of clopen up-sets of and each closed down-set of is an intersection of clopen down-sets of .

::(d) Clopen up-sets and clopen down-sets of form a subbasis for .

::(e) For each pair of closed subsets and of , if , then there exists a clopen up-set such that and .

A Priestley morphism from a Priestley space to another Priestley space is a map which is continuous