Equivalence to the axiom of choice of spatiality results in Section 4

Determine whether any of the following spatiality results are equivalent to the axiom of choice (AC): (i) every compact subfit frame is spatial (Isbell’s Spatiality Theorem); (ii) every compact T1 McKinsey–Tarski (MT) algebra is spatial (Nöbeling’s Spatiality Theorem); (iii) every N-locally compact T1 MT-algebra is spatial (Nöbeling’s Spatiality Theorem); and (iv) every locally compact Tδ MT-algebra is spatial (Theorem thm: td spatial).

Background

Section 4 develops connections between spatiality results in pointfree topology and the axiom of choice (AC). The authors prove that several compactness-related properties in MT-algebras are equivalent to AC and use AC to derive spatiality theorems, including Nöbeling’s Spatiality Theorems and Isbell’s Spatiality Theorem.

Despite these results, the exact logical strength of the spatiality theorems relative to AC is unresolved. Specifically, it is unknown whether the spatiality theorems themselves are equivalent to AC, i.e., whether AC is necessary and sufficient for each of these spatiality assertions.