A De Vries-type Duality Theorem for Locally Compact Spaces -- I

A duality theorem for the category of locally compact Hausdorff spaces and continuous maps which generalizes the well-known Duality Theorem of de Vries is proved.

💡 Research Summary

The paper establishes a categorical duality between locally compact Hausdorff spaces (LCH) with continuous maps and a newly introduced algebraic structure called a local contact algebra (LCA). This result generalizes de Vries’s classic duality, which links compact Hausdorff spaces to Boolean algebras equipped with a contact relation. The authors begin by reviewing the limitations of the original de Vries framework when applied to non‑compact, locally compact spaces: the global compactness that underlies the construction of regular closed sets and the contact relation is no longer available, so a Boolean algebra alone cannot capture the full topological information.

To overcome this, they define an LCA as a triple (B, C, I) where B is a complete Boolean algebra, C ⊆ B × B is a de Vries‑type contact relation (reflexive, symmetric, and satisfying the usual approximation axioms), and I is an ideal of B that represents the collection of regular open sets of the underlying space. The ideal I isolates the “local” part of the algebra: elements of B \ I correspond to compact regular closed subsets, while elements of I encode the non‑compact open structure. This separation mirrors the way a locally compact space can be viewed as a union of its compact neighborhoods.

Morphisms between LCAs, called local contact morphisms, are Boolean homomorphisms f : B → B′ that preserve the ideal (f