Local compactness does not always imply spatiality

TL;DR

The paper demonstrates that local compactness does not guarantee spatiality in MT-algebras, providing a counterexample, and explores the relationship between N"obeling's separation axioms and spatiality in pointfree topology.

Contribution

It constructs a counterexample of a locally compact sober MT-algebra that is not spatial and connects N"obeling's separation axioms to spatiality results in pointfree topology.

Findings

Counterexample of a non-spatial locally compact sober MT-algebra

N"obeling's separation axioms relate closely to MT-algebra properties

Every locally compact T_{1/2}-algebra is spatial, assuming the axiom of choice

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.