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and order-preserving.

Let Pries denote the category of Priestley spaces and Priestley morphisms.

Connection with spectral spaces

Priestley spaces are closely related to spectral spaces. For a Priestley space , let denote the collection of all open up-sets of . Similarly, let denote the collection of all open down-sets of .

Theorem:If is a Priestley space, then both and are spectral spaces.

Conversely, given

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