A new duality theorem for locally compact spaces

In 1962, H. de Vries proved a duality theorem for the category {\bf HC} of compact Hausdorff spaces and continuous maps. The composition of the morphisms of the dual category obtained by him differs from the set-theoretic one. Here we obtain a new category dual to the category {\bf HLC} of locally compact Hausdorff spaces and continuous maps for which the composition of the morphisms is a natural one but the morphisms are multi-valued maps.

💡 Research Summary

The paper revisits H. de Vries’s 1962 duality theorem, which establishes an equivalence between the category HC of compact Hausdorff spaces with continuous maps and a category of contact algebras equipped with “continuous contact‑preserving” morphisms. Although mathematically correct, de Vries’s construction has a notable drawback: the composition of morphisms in the dual category does not coincide with the ordinary set‑theoretic composition of functions, making the categorical picture less natural and harder to work with in applications.

The authors aim to overcome this limitation by developing a new duality for the broader category HLC of locally compact Hausdorff spaces and continuous maps. Their strategy is to replace the single‑valued, composition‑complicated morphisms of de Vries with multivalued maps that preserve contact and regularity, thereby restoring the usual composition law.

Key technical steps are as follows:

1. Local Contact Algebras (LCAs).
   For each locally compact Hausdorff space (X), the authors consider the Boolean lattice (\operatorname{RO}(X)) of regular open subsets. They equip (\operatorname{RO}(X)) with a contact relation (C_X) defined by non‑empty intersection: (U,C_X,V) iff (U\cap V\neq\varnothing). Together, ((\operatorname{RO}(X),C_X)) forms a local contact algebra, a natural extension of de Vries’s contact algebras that accommodates non‑compact spaces. Points of (X) correspond to filters (\mathcal{F}_x={U\in\operatorname{RO}(X)\mid x\in U}) in this algebra, providing an algebraic representation of the underlying topology.

2. Multivalued Morphisms.
   A morphism (f\colon X\to Y) in HLC is represented dually by a multivalued map (\widehat{f}\colon\operatorname{RO}(X)\rightrightarrows\operatorname{RO}(Y)) satisfying two conditions:

   • Contact Preservation: If (U,C_X,V) then for every (U’\in\widehat{f}(U)) and (V’\in\widehat{f}(V)) we have (U’,C_Y,V’).
   • Regularity Preservation: Each image (\widehat{f}(U)) consists solely of regular open sets, and the union (\bigcup\widehat{f}(U)) is again regular open.
     These requirements guarantee that (\widehat{f}) respects the topological structure encoded in the LCAs.

3. Natural Composition.
   Given (\widehat{f}\colon\operatorname{RO}(X)\rightrightarrows\operatorname{RO}(Y)) and (\widehat{g}\colon\operatorname{RO}(Y)\rightrightarrows\operatorname{RO}(Z)), the composition is defined pointwise by
   \