On the Duality of Compact vs. Open

Abstract

ABSTRACT: It is a pleasant fact that Stone‐duality may be described very smoothly when restricted to the category of compact spectral spaces: The Stone‐duals of these spaces, arithmetic algebraic lattices, may be replaced by their sublattices of compact elements thus discarding infinitary operations. We present a similar approach to describe the Stone‐duals of coherent spaces, thus dropping the requirement of having a base of compact‐opens (or, alternatively, replacing algebraicity of the lattices by continuity). The construction via strong proximity lattices is resembling the classical case, just replacing the order by an order of approximation . Our development enlightens the fact that “open” and “compact” are dual concepts which merely happen to coincide in the classical case.