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Preliminaries A partially ordered space is both a poset and a topological space. It has connected components both as a topological space, and connected components as a poset, i.e. the maximal connected subposets. These poset components have less global structure among each other than the topological components, i.e. we have

Any poset can be written as a disjoint union of its components.

However, even for a Priestley space, the poset components are not necessarily closed.


I would need to define a generalized connected components decomposition for Priestley spaces equivalent to the one which would be constructed by the following "transfinite" procedure:

  1. Start with the decomposition of the Priestley space into its poset components.
  2. Take to (topological) closure of the components of the current decomposition.
  3. Aggregate the components which overlap after the closure from step 2. into coarser components, forming a new decomposition.
  4. Go to step 2. and iterate the procedure transfinitely, until we have a decomposition into closed components.

I'm not familiar enough with transfinite induction to be sure whether this procedure is really well defined. It's easy to construct a Priestley space from an ordinal space such that this procedure really has to be iterated transfinitely up to any prescribed ordinal number before it terminates. I wonder whether there isn't a definition of this generalized connected components decomposition which avoids the questionable transfinite induction.


Edit I finally found a counter-example to the question that initially interested me, and that motivated me to learn all this stuff about Priestley spaces. However, I decided not to delete this question, because it is still a valid question about (avoiding) transfinite induction.

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The transfinite process that you mentioned above is well defined since there is a limitless supply of ordinals and you never enter an infinite loop in the transfinite process. On the other hand, one can avoid transfinite induction simply by using connectedness in a different topology. Let $X$ be a poset. We say that $U\subseteq X$ is an upper set if whenever $x\leq y$ and $x\in U$, then $y\in U$ as well. The Alexandroff topology on $X$ is the topology where the open sets are precisely the upper sets. It is easy to see that a poset $X$ is connected as a poset if and only if $X$ is connected in the Alexandroff topology.

Now let $X$ be a Priestley space. Let $\mathcal{T}$ be the topology on $X$ where $U\in\mathcal{T}$ if and only if $U\subseteq X$ is an open upper set. Then what you call a component in a Priestley space should be the components in the topology $\mathcal{T}$ (I have not checked all the details in making sure that this notion of connectedness coincides with the transfinite procedure you mentioned yet but it seems like it will work.).

I should also mention that the topology $\mathcal{T}$ mentioned above has uses besides defining a notion of connectedness for Priestley spaces. The topology $(X,\mathcal{T})$ turns out to be a stably compact space. Furthermore, one obtains a duality between all stably compact spaces and compact Hausdorff ordered spaces $X$ (where $\leq$ is closed in $X^{2}$) using the topology consisting of upper open sets.

These notes give some information on the duality between stably compact spaces and compact Hausdorff ordered spaces.

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