Continuous separation of fields and their complements

Determine whether every field of real numbers and the complement in the real line of every such field is a CT3-space.

Background

The paper introduces CTi-spaces by requiring the existence of continuous separation maps on the hyperspace of nonempty closed sets equipped with the Vietoris topology. It proves that every field of real numbers and its complement spaces are CT2 but not CT4. The authors leave unresolved whether these spaces satisfy the stronger CT3 axiom.

References

All fields of real numbers and their complement spaces are CT2 but not CT4; however, we do not know whether they are CT3.

— Axioms of Continuous Separation  (2608.13086 - Yang et al., 13 Aug 2026) in Abstract

Whether or not the following facts hold: $CT_3 \ni ,$?

— Axioms of Continuous Separation  (2608.13086 - Yang et al., 13 Aug 2026) in Section 6, Problem