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Functional Fraïssé’s Conjecture (FFC) for scattered functions

Establish whether continuous reducibility forms a better-quasi-order (bqo) on the class of scattered functions (i.e., functions f such that every non-empty subset of the domain contains a non-empty open set on which f is constant).

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Background

The paper proves that continuous reducibility is a bqo on broad classes of continuous scattered functions with zero-dimensional separable metrizable domains and metrizable codomains. This conjecture asks for a significant generalization: extending the bqo result to the entire class of scattered functions, without restricting to continuity or specific domain/codomain classes. It echoes Fraïssé's conjecture for scattered linear orders, seeking a similarly comprehensive bqo phenomenon for functions.

References

In this respect it does make sense to conjecture that scattered functions (even discontinuous ones) may be constructed from simpler ones, which might in turn allow for a (very general) bqo result, reminiscent of Fraïssé's Conjecture for scattered linear orders.

Conjecture [Functional Fraïssé's Conjecture - FFC] Continuous reducibility is bqo on the class of scattered functions.

— A well-quasi-order for continuous functions (2410.13150 - Carroy et al., 17 Oct 2024) in Section 7.2 (Sharpness)