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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?

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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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    $\begingroup$ A note -- if you're linking to arXiv, it's better to link to the abstract (arxiv.org/abs/0705.4259) rather than directly to the PDF. From the abstract, one can easily click through to the PDF; not so the reverse. And the abstract allows one to do things like see different versions of the paper, search for other things by the same authors, etc. Thank you! $\endgroup$
    – Harry Altman
    Commented May 4, 2015 at 10:50
  • $\begingroup$ Thanks @HarryAltman , this makes sense, and I edited my post accordingly. $\endgroup$
    – Dominic van der Zypen
    Commented May 4, 2015 at 10:55

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