Local compactness does not always imply spatiality

Ponente(s): Ranjitha Raviprakash Amrutha

We revisit N\"obeling's largely overlooked work on pointfree separation axioms and local compactness from the 1950s and compare them with their modern formulations for MT-algebras. Most axioms agree, though N\"obeling's versions of Hausdorffness and local compactness are weaker. His spatiality theorem, that compact $T_1$-algebras are spatial, yields Isbell's theorem that compact subfit frames are spatial, and we extend it by showing that locally compact $T_{1/2}$-algebras are spatial. Since frames correspond to $T_{1/2}$-algebras, this sheds light on why locally compact frames are spatial. Finally, we construct a locally compact sober MT-algebra that is not spatial, showing that local compactness does not guarantee spatiality for MT-algebras.