The Fell Compactification of a Poset
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A poset P forms a locally compact \(T_0\)-space in its Alexandroff topology. We consider the hit-or-miss topology on the closed sets of P and the associated Fell compactification of P. We show that the closed sets of P with the hit-or-miss topology is the Priestley space of the bounded distributive lattice freely generated by the order dual of P. The Fell compactification of H(P) is shown to be the Priestley space of a sublattice of the upsets of P consisting of what we call Fell upsets. These are upsets that are finite unions of those obtained as upper bounds of finite subsets of P. The restriction of the hit topology to H(P) is a stable compactification of P. When P is a chain, we show that this is the least stable compactification of P.
KeywordsHit-or-miss topology Fell compactification Stable compactification Order-compactification Poset Free lattice Distributive lattice Priestley space
MSC:54B20 54D35 54D45 06E15 06B25
We would like to thank Vladik Kreinovich for the opportunity to contribute to this volume.
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