Topological aspects of poset spaces

We study two classes of spaces whose points are filters on partially ordered sets. Points in MF spaces are maximal filters, while points in UF spaces are unbounded filters. We give a thorough account of the topological properties of these spaces. We obtain a complete characterization of the class of countably based MF spaces: they are precisely the second-countable T_1 spaces with the strong Choquet property. We apply this characterization to domain theory to characterize the class of second-countable spaces with a domain representation.

💡 Research Summary

The paper investigates two families of topological spaces that arise from filters on partially ordered sets (posets). A filter on a poset P is a non‑empty upward‑closed set closed under finite meets. The authors focus on two distinguished subclasses: maximal filters (MF) – filters that cannot be properly extended – and unbounded filters (UF) – filters that have no upper bound in P. By taking each filter as a point and declaring the basic open sets to be “all filters containing a given element p∈P”, they obtain the MF‑space and the UF‑space associated with P.

The first part of the paper establishes basic separation and countability properties. MF‑spaces are always T₁ because for any two distinct maximal filters one can find an element of the poset that belongs to one filter but not the other, yielding a separating basic open set. UF‑spaces need not be T₁; indeed, the lack of maximality often prevents point‑distinguishing open sets. Both constructions inherit the order‑theoretic structure of P, and the authors show that first‑countability is equivalent to the existence of a countable cofinal subset of P.

The central contribution is a complete characterization of second‑countable MF‑spaces. The authors prove the following theorem: a space X is homeomorphic to a second‑countable MF‑space if and only if X is second‑countable, T₁, and possesses the strong Choquet property. The strong Choquet property is defined via a two‑player game in which Player I chooses a decreasing sequence of non‑empty open sets and Player II selects points inside them; Player II has a winning strategy exactly when the space is strong Choquet. The proof proceeds by constructing a countable basis of the poset from a winning strategy and showing that the resulting MF‑space reproduces the original topology. Conversely, any second‑countable MF‑space admits a natural winning strategy for Player II, establishing the equivalence.

Having identified the precise topological signature of MF‑spaces, the authors turn to domain theory. A domain representation of a topological space is a continuous dcpo (directed‑complete partial order) equipped with the Scott topology such that the space is homeomorphic to the set of maximal elements of the domain. The paper shows that a second‑countable space admits a domain representation exactly when it is a second‑countable MF‑space. In other words, the class of second‑countable spaces with a domain representation coincides with the class characterized by the strong Choquet property and T₁ separation. This bridges the gap between order‑theoretic domain models (Scott/Lawson topologies) and classical descriptive set‑theoretic properties.

The final section examines UF‑spaces. The authors demonstrate that UF‑spaces generally fail to be Baire, fail to be strong Choquet, and may lack even T₁ separation. They provide explicit examples where a countable poset yields a UF‑space that is not second‑countable or not Hausdorff, underscoring the essential role of maximality in obtaining nice topological behavior.

Overall, the paper contributes a clean and elegant taxonomy: MF‑spaces are precisely the second‑countable T₁ strong Choquet spaces, and these are exactly the second‑countable spaces that can be represented by continuous domains. This result unifies several strands of topology, order theory, and theoretical computer science, and opens avenues for further research on generalized filter spaces, game‑theoretic characterizations, and applications to semantics of computation.