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An overlooked weakening of perfect normality: Perfect regularity in spaces and locales

  • Ana Belén Avilez
  • Guram Bezhanishvili
  • Joanne Walters-Wayland
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Abstract

We introduce the notion of perfect regularity as an appropriate weakening of perfect normality, both for spaces and locales. Various characterizations are given, using Dedekind-MacNeille completions, injective hulls, and sublocales. We place the new class of perfectly regular frames among various well-studied classes of frames. We also introduce the construction of perfect regularization of a completely regular frame, compare it to Isbell's well-known booleanization construction, and argue that it is at least as important as the latter.

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Cite

BibTeXRIS
BibTeX
@misc{avilez2026overlooked,
  title = {{An overlooked weakening of perfect normality: Perfect regularity in spaces and locales}},
  author = {Ana Belén Avilez and Guram Bezhanishvili and Joanne Walters-Wayland},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.06990},
  eprint = {2609.06990},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.06990},
}
RIS
TY  - UNPB
TI  - An overlooked weakening of perfect normality: Perfect regularity in spaces and locales
AU  - Ana Belén Avilez
AU  - Guram Bezhanishvili
AU  - Joanne Walters-Wayland
PY  - 2026
DA  - 2026/09/07
PB  - arXiv
DO  - 10.48550/arxiv.2609.06990
UR  - https://arxiv.org/abs/2609.06990
AB  - We introduce the notion of perfect regularity as an appropriate weakening of perfect normality, both for spaces and locales. Various characterizations are given, using Dedekind-MacNeille completions, injective hulls, and sublocales. We place the new class of perfectly regular frames among various well-studied classes of frames. We also introduce the construction of perfect regularization of a completely regular frame, compare it to Isbell's well-known booleanization construction, and argue that it is at least as important as the latter.
ER  -