Dugundji系统与有效零维的收缩刻画
Dugundji systems and a retract characterization of effective zero-dimensionality
摘要：
在本文中（如[Ken 15]），我们考虑了可分度量空间为零维的一个有效刻画当且仅当每个非空闭子集都是该空间的一个收缩集（实际上，这是我们已经证明的关于固定空间的零维闭子空间的一个相对结果）。这使用（在相反的方向）局部紧性和双定位集[Ken 15]，但在向前的方向，新版本有一个更简单的证明，没有紧性假设。此外，证明的前向蕴涵涉及到所谓的Dugundji系统：我们阐述了这样的系统的一般建设一个适当的非空闭子空间（使用可数仿紧性的可计算形式），和修改-使集成对不相交，如果子空间是零维的，或避免限制适当的子空间。在另一个方向上，第二个定理在$p$ -进分析中应用第一个定理的思想来计算更一般形式的收缩，给定一个Dugundji系统（可能没有不相交性）。最后，我们通过改进等价蕴涵，补充了上述零维子空间的有效收缩刻画（或在某些情况下Weihrauch约简），对于具有某些负信息的至多零维闭子集，在[Ken 15，§4]中引入的运算$N,M,B,S$的可计算性的单独条件中，对应于消失的大归纳维数，消失的小归纳维数，相对闭集的可数基的存在性和开集序列的约化原理。因此，类似于[Ken 15]中可计算度量空间的有效零维的鲁棒概念，对于由至多零维闭子集组成的表示点类，存在“一致有效零维”的鲁棒概念。
英文摘要：
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,§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.
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