Dugundji systems and a retract characterization of effective zero-dimensionality

Abstract

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.