Sober Spaces, Well-Filtration and Compactness Principles

We investigate various notions of sobriety for topological spaces that are equivalent in ZF set theory with the Axiom of Choice but may differ in the absence of choice principles. The Ultrafilter Principle UP (alias Prime Ideal Theorem) suffices for the desired conclusions. We derive from UP three topological postulates and prove their equivalence in ZF without choice: the Well-Filtration Principle, the Compactness Principle and the Irreducible Cutset Principle. On the other hand, the latter, applied to A(lexandroff)-spaces, immediately yields Rudin’s Lemma, while the other two, restricted to A-spaces, are equivalent to the statement that for any Noetherian poset, the ∧-semilattice of all finitely generated upper sets is again Noetherian. Mathematics Subject Classification: Primary: 03E25. Secondary: 06D20, 54D10, 54D30, 54D45.