Dugundji systems and a retract characterization of effective zero-dimensionality
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In a previous paper, the author considered several conditions for effective zero-dimensionality of a computable metric space X; each of the (classically equivalent) properties of having vanishing small or large inductive dimension, or covering dimension, or having a countable basis of clopen sets, can be interpreted as multi-valued operations, and the computability of these operations was shown to be mutually equivalent. If one fixes a represented set of at-most-zero-dimensional subspaces, suitable uniform versions of some of the implications in the equivalence also hold. In this paper, we show for closed at-most-zero-dimensional subspaces with some negative information, we again have equivalence of computability of the operations considered (thus there is a robust notion of `uniform effective zero-dimensionality' for such classes of subsets). We also extend a retract characterization of effective zero-dimensionality from our earlier paper (recall a space is zero-dimensional iff every nonempty closed subset of it is a retract of the space) to closed subspaces. While, in the converse direction, local compactness & bilocated sets are still used, in the forward direction the new, relative version of this result has a simpler proof (even when the subspace coincides with the whole space) and no compactness assumption. Actually, this proof 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, our 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).
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