# Priestley topologizability and connected components

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?

## 1 Answer

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.

• 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! May 4 '15 at 10:50
• Thanks @HarryAltman , this makes sense, and I edited my post accordingly. May 4 '15 at 10:55