A duality for the class of compact T1 -topological spaces

We present a duality between compact \( T_1 \)-spaces and a class of distributive lattices (subfit, compact, and complete), which captures key aspects of both Stone duality and \(\Omega\)-point duality in particular instances. We then show how this duality extends to a contravariant adjunction between \( T_1 \)-spaces and bounded distributive lattices. This adjunction gives rise to a canonical compactification---the \emph{Wallman compactification}---for \( T_1 \) spaces, such that any \textit{strongly continuous} map from a \( T_1 \) space \( X \) into a compact \( T_1 \) space factors uniquely through the Wallman compactification of \( X \). This is joint work with Matteo Viale and Mai Gehrke.