TITLE: Priestley topologizability and connected components
QUESTION [0 upvotes]: This question is in the spirit of another older question.
We say that a poset $(P,\leq)$ is Priestley-topologizable if there is a topology $\tau$ on $P$ such that $(P,\leq,\tau)$ is a Priestley space.
Is it true that a poset $(P,\leq)$ is Priestley-topologizable if and only if all its connected components are Priestley-topologizable?

REPLY [1 votes]: Only one implication of the equivalence appearing in the question holds.
It is true that if each connected component of a poset $(P,\tau)$ can be endowed with a Priestley topology, then the same holds for the whole poset. There is a construction in the proof of Lemma 2.3 of this paper. 
However, in the same paper in Lemma 2.4, a construction of a Priestley space $(P,\leq,\tau)$ is given such that one connected component of $(P,\leq)$ is not Priestley-topologizable.
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