Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. 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 between the category of Priestley spaces and the category of bounded distributive lattices. A Priestley space is an ordered topological space (X, ,<=), i. e. a set X equipped with a partial order <= and a topology , satisfying the following two conditions: (i) (X, ) is compact. (ii) If scriptstyle x,notle, 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.). Priestley spaces are closely related to spectral spaces. For a Priestley space (X, ,<=), let u denote the collection of all open up-sets of X. Similarly, let d denote the collection of all open down-sets of X.

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