This question is in the spirit of [another older question](https://mathoverflow.net/questions/142540/are-the-connected-components-of-a-priestley-space-closed/).


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](http://en.wikipedia.org/wiki/Priestley_space).


Is it true that a poset $(P,\leq)$ is Priestley-topologizable if and only if all its connected components are Priestley-topologizable?



Priestley topologizability and connected components
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](http://arxiv.org/abs/0705.4259). 


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.


