This question is in the spirit of [another older question][1].

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][2].

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


  [1]: http://mathoverflow.net/questions/142540/are-the-connected-components-of-a-priestley-space-closed/
  [2]: http://en.wikipedia.org/wiki/Priestley_space