In this paper (as in [Ken15]), we consider an effective version of the characterization of separable metric spaces as zero-dimensional iff every nonempty closed subset is a retract of the space (actually, it is a relative result for closed zero-dimensional subspaces of a fixed space that we have proved). This uses (in the converse direction) local compactness & bilocated sets as in [Ken15], but in the forward direction the newer version has a simpler proof and no compactness assumption. Furthermore, the proof of the forward implication relates to so-called Dugundji systems: we elaborate both a general construction of such systems for a proper nonempty closed subspace (using a computable form of countable paracompactness), and modifications -- to make the sets pairwise disjoint if the subspace is zero-dimensional, or to avoid the restriction to proper subspaces. In a different direction, a second theorem applies in $p$-adic analysis the ideas of the first theorem to compute a more general form of retraction, given a Dugundji system (possibly without disjointness). Finally, we complement the effective retract characterization of zero-dimensional subspaces mentioned above by improving to equivalence the implications (or Weihrauch reductions in some cases), for closed at-most-zero-dimensional subsets with some negative information, among separate conditions of computability of operations $N,M,B,S$ introduced in [Ken15,\S 4] and corresponding to vanishing large inductive dimension, vanishing small inductive dimension, existence of a countable basis of relatively clopen sets, and the reduction principle for sequences of open sets. Thus, similarly to the robust notion of effective zero-dimensionality of computable metric spaces in [Ken15], there is a robust notion of `uniform effective zero-dimensionality' for a represented pointclass consisting of at-most-zero-dimensional closed subsets.

翻译：本文( 如 [ Ken15] ) 中, 我们考虑一个有效的版本, 将可分离的计量空间定性为零维度, 如果每个非空的封闭子集都是空间的缩回( 事实上, 这是我们所证明的固定空间封闭的零维子空间的相对结果 ) 。 这使用( 反方向 ) 本地缩缩缩和双位集组, 如 [ Ken15], 但在前方方向, 新版本有一个更简单的证明, 没有缩缩缩假设。 此外, 远隐的证明与所谓的 Dugondji 系统有关: 我们为适当的非空闭闭的子空间进行总体构造( 使用可量化的可计算形式), 修改 -- 如果子空格为零维度, 或者避免对合适的子空间的限制。 在不同的方向, 以美元为第二位, 直角分析第一个理论的理念, 将一个更普遍的缩缩略式的缩放形式, 以直径直径的直径直径直径的运行方式, 在最后的直径直径直径直立的直径直立的平基的直径直径基系统上, 。