Equilogical spaces form a cartesian closed complete category that contains all T0 spaces. Assemblies (on algebraic lattices) are a slight generalisation of an equivalent description of this category. They too form a cartesian closed complete category that even contains all topological spaces. We introduce various notions of completeness for assemblies, some inspired by Synthetic Domain Theory, and others inspired by topology, e.g., sobriety. Finally, we introduce the class of ordered assemblies that still contains all topological spaces, but excludes certain pathologies.
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Reinhold Heckmann /