Preservation of strong countable-dimensionality and zero-dimensionality under uniformly continuous surjections Cp(X) -> Cp(Y)

Determine whether, for a uniformly continuous surjection T : Cp(X) -> Cp(Y) with X a compact metrizable strongly countable-dimensional space (respectively, a compact metrizable zero-dimensional space), the space Y is necessarily strongly countable-dimensional (respectively, zero-dimensional).

Background

The paper studies uniformly continuous surjections between pointwise-topologized function spaces Cp(X) and Cp(Y) over metrizable spaces. While many invariance results are known for uniform homeomorphisms and linear surjections, substantially less is established for mere uniformly continuous surjections. Prior work showed positive results under stronger hypotheses, such as c-good maps or σ-compactness, but the general case remained unresolved.

Problem 1.2 from [13] asks whether zero-dimensionality and strong countable-dimensionality are preserved under uniformly continuous surjections Cp(X) -> Cp(Y) when X is compact metrizable. The present paper proves preservation under the weaker assumption that T is uniformly continuous and inversely bounded, and for all metrizable (not necessarily compact) X, thereby strengthening previous results but not fully resolving the original question without the inverse boundedness condition.