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.