I believe using A4 (and maybe also A5) in multiple places will be important to proving a positive result. This is because A1, A2, and A3 are extremely week on their own.
A1-A3 is not even enough to prove C1. To see a counterexample, take any well ordering on R/Q, and consider the preference ordering over the space of lotteries on a two element set of deterministic outcomes. If two lotteries have probabilities of the first outcome that differ by a rational number, they are equivalent, otherwise, you compare them according to your well ordering. This clearly satisfies A1 and A2, and it satisfies A3, since every nonempty open set contains lotteries incomparable with any given lottery. However, has a continuum length ascending chain of strict preference, and so cant be captured in a function to the interval.
Further, one might hope that C1 together with A3 would be enough to conclude C2, but this is also not possible, since there are discontinuous functions on the simplex that are continuous when restricted to any line segment in the domain.
In both of these cases, it seems to me like there is hope that A4 provides enough structure to eliminate the pathological counterexamples, since there is much less you can do with convex upsets.
I believe using A4 (and maybe also A5) in multiple places will be important to proving a positive result. This is because A1, A2, and A3 are extremely week on their own.
A1-A3 is not even enough to prove C1. To see a counterexample, take any well ordering on R/Q, and consider the preference ordering over the space of lotteries on a two element set of deterministic outcomes. If two lotteries have probabilities of the first outcome that differ by a rational number, they are equivalent, otherwise, you compare them according to your well ordering. This clearly satisfies A1 and A2, and it satisfies A3, since every nonempty open set contains lotteries incomparable with any given lottery. However, has a continuum length ascending chain of strict preference, and so cant be captured in a function to the interval.
Further, one might hope that C1 together with A3 would be enough to conclude C2, but this is also not possible, since there are discontinuous functions on the simplex that are continuous when restricted to any line segment in the domain.
In both of these cases, it seems to me like there is hope that A4 provides enough structure to eliminate the pathological counterexamples, since there is much less you can do with convex upsets.