Local compactness does not always imply spatiality

Abstract

It is a well-known result in pointfree topology that every locally compact frame is spatial. Whether this result extends to MT-algebras (McKinsey-Tarski algebras) was an open problem. We resolve it in the negative by constructing a locally compact sober MT-algebra which is not spatial. We also revisit N\"obeling's largely overlooked approach to pointfree topology from the 1950s. We show that his separation axioms are closely related to those in the theory of MT-algebras with the notable exception of Hausdorffness. We prove that N\"obeling's Spatiality Theorem implies the well-known Isbell Spatiality Theorem. We then generalize N\"obeling's Spatiality Theorem by proving that each locally compact $T_{1/2}$-algebra is spatial. The proof utilizes the fact that every nontrivial $T_{1/2}$-algebra contains a closed atom, which we show is equivalent to the axiom of choice.