Scatteredness and strongly σ-scatteredness under uniformly continuous surjections

Ascertain whether, for metrizable spaces X and Y, the conclusions that Y is scattered whenever X is scattered and that Y is strongly σ-scattered whenever X is strongly σ-scattered remain valid when T : Dp(X) -> Dp(Y) is assumed only uniformly continuous and surjective (without linearity), where Dp(X) denotes either Cp(X) or C*(X) with the pointwise convergence topology.

Background

The paper proves that scatteredness and strongly σ-scatteredness are preserved under linear continuous surjections between Dp(X) spaces over metrizable spaces (Theorem 4.2 and Proposition 4.3).

The authors explicitly state that it is unknown whether these results extend to the setting where T is merely uniformly continuous and surjective. They indicate that progress on this question is tied to a separate major open problem concerning preservation of complete metrizability under uniform surjections between function spaces.

References

We don't know whether analogues of Theorem 4.2 and Proposition 4.3 are valid under a weaker assumption: T : Dp(X) -> Dp(Y) is a uniformly con- tinuous surjection. This is because the following major question posed by Marciszewski and Pelant is open.

— On uniformly continuous surjections between $C_p$-spaces over metrizable spaces (2408.01870 - Eysen et al., 3 Aug 2024) in Section 4, preceding Problem 4.4