bibkey: banaschewski1955nulldimensional
authors: Bernhard Banaschewski
year: 1955
title: Über nulldimensionale Räume
doi: 10.1002/mana.19550130302
url: https://doi.org/10.1002/mana.19550130302
claim: The zero-dimensional compactification underlying the boundary-avoiding Boolean graph construction is classically attributed to Stone and Banaschewski.
strata_touched:
  - D5/S3/Topology/Resolution
  - D5/S1/Dynamics/StableRange
license: citation-only
triage: anchor

Classical attribution and exact formalized scope

Verified locator

DOI: 10.1002/mana.19550130302. Canonical URL: https://doi.org/10.1002/mana.19550130302. Crossref verifies the bibliographic metadata only; the original article has not been inspected.

Construction and scope

Banaschewski’s article appears in Mathematische Nachrichten 13, 129–140. The title, author, bibliographic coordinates and DOI are verified against Crossref. The original article has not been inspected, so no theorem number or claim about its exact relative formulation is attributed to it.

For compact Hausdorff X and closed zero-dimensional C, consider all actual subsets A of X whose ambient frontiers miss C. Their characteristic functions are locally constant near each point of C. They are the clopen subsets of the Michael modification obtained by isolating X outside C while retaining the original neighborhoods at C. The classical Stone/Banaschewski picture motivates compactification by all these Boolean coordinates.

RelativeBoolean formalizes the closure of the actual membership graph in X times the binary cube. It directly proves the separating-neighborhood lemma, injection of the cube projection, zero-dimensionality, singleton fibers over C, and a continuous section homeomorphic to the full inverse image of C. Continuous maps of pairs act by actual inverse images of admissible subsets, giving continuous coherent lifts. Identification with a universal compactification is explanatory context, not an additional formalized universal-property theorem.

The relative graph arguments and the stable-range identity are presented as repository derivations using pinned Mathlib separation and compactness results. The compactification itself is classical. There is no mathematical priority claim. No metric, countability, nonemptiness, openness or onto condition on pair maps is imposed. The stable-range identity uses all natural iterates including zero and needs only continuity on a topological space. Compact Hausdorff hypotheses separately give a compact stable core with an onto continuous restriction.

These results do not assert shadowing for the resolution. They do not supply finite observation names, a core period or a general shadowing extension. Peng’s unrestricted Question 5.1 remains outside the construction.