Generalized connected components decomposition for Priestley spaces 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:


*

*Start with the decomposition of the Priestley space into its poset components.

*Take to (topological) closure of the components of the current decomposition.

*Aggregate the components which overlap after the closure from step 2. into coarser components, forming a new decomposition.

*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.
 A: 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.
