Locally Compact Stone Duality

Abstract

We prove a number of dualities between posets and (pseudo)bases of open sets in locally compact Hausdorff spaces. In particular, we show that (1) Relatively compact basic sublattices are finitely axiomatizable. (2) Relatively compact basic subsemilattices are those omitting certain types. (3) Compact clopen pseudobasic posets are characterized by separativity. We also show how to obtain the tight spectrum of a poset as the Stone space of a generalized Boolean algebra that is universal for tight representations.

Introduction reorganized and some references added

References in corpus (2)

• Stone duality above dimension zero: Axiomatising the algebraic theory of C(X)
• Model theory of ${\mathrm{C}}^{\ast }$-algebras

Cited by in corpus (2)

• General Non-Commutative Locally Compact Locally Hausdorff Stone Duality
• A general Banach-Stone type theorem and applications