Staš, Michal The regularity properties on the real line. (English) Zbl 1262.03099 Acta Univ. Carol., Math. Phys. 51, Suppl., 73-82 (2010). The author discusses these regularity properties along with other related properties of the real line (such as existence of a Bernstein set, existence of a selector for Lebesgue decomposition etc.) in different theories. While in ZF these three properties are not provable, in ZFC they are false, and J. Mycielski has proven that in \(\mathrm{ZF}+\mathrm{AD}\) these properties of the real line are true. The author discusses these properties under different axioms, such as wAC (weak form of AC), existence of free ultrafilters, \(\mathrm{AC}_{2}\) etc. Reviewer: Gowri Navada (Salem) MSC: 03E10 Ordinal and cardinal numbers 03E25 Axiom of choice and related propositions 03E35 Consistency and independence results 03E75 Applications of set theory Keywords:real line; Lebesgue measurability; Baire property; perfect sets; axiom of choice; consistency; regularity × Cite Format Result Cite Review PDF Full Text: Link