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Pervin spaces and Frith frames: bitopological aspects and completion

Célia Borlido, Anna Laura Suarez

Abstract

A Pervin space is a set equipped with a bounded sublattice of its powerset, while its pointfree version, called Frith frame, consists of a frame equipped with a generating bounded sublattice. It is known that the dual adjunction between topological spaces and frames extends to a dual adjunction between Pervin spaces and Frith frames, and that the latter may be seen as representatives of certain quasi-uniform structures. As such, they have an underlying bitopological structure and inherit a natural notion of completion. In this paper we start by exploring the bitopological nature of Pervin spaces and of Frith frames, proving some categorical equivalences involving zero-dimensional structures. We then provide a conceptual proof of a duality between the categories of $T_0$ complete Pervin spaces and of complete Frith frames. This enables us to interpret several Stone-type dualities as a restriction of the dual adjunction between Pervin spaces and Frith frames along full subcategory embeddings. Finally, we provide analogues of Banaschewski and Pultr's characterizations of sober and $T_D$ topological spaces in the setting of Pervin spaces and of Frith frames, highlighting the parallelism between the two notions.

Pervin spaces and Frith frames: bitopological aspects and completion

Abstract

A Pervin space is a set equipped with a bounded sublattice of its powerset, while its pointfree version, called Frith frame, consists of a frame equipped with a generating bounded sublattice. It is known that the dual adjunction between topological spaces and frames extends to a dual adjunction between Pervin spaces and Frith frames, and that the latter may be seen as representatives of certain quasi-uniform structures. As such, they have an underlying bitopological structure and inherit a natural notion of completion. In this paper we start by exploring the bitopological nature of Pervin spaces and of Frith frames, proving some categorical equivalences involving zero-dimensional structures. We then provide a conceptual proof of a duality between the categories of complete Pervin spaces and of complete Frith frames. This enables us to interpret several Stone-type dualities as a restriction of the dual adjunction between Pervin spaces and Frith frames along full subcategory embeddings. Finally, we provide analogues of Banaschewski and Pultr's characterizations of sober and topological spaces in the setting of Pervin spaces and of Frith frames, highlighting the parallelism between the two notions.
Paper Structure (24 sections, 60 theorems, 26 equations)

This paper contains 24 sections, 60 theorems, 26 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Basic notation
  4. Bitopological spaces and biframes
  5. The Prime Ideal Theorem
  6. Pervin spaces and Frith frames
  7. Symmetrization
  8. Forgetful functors
  9. Open intersections and strongly exact meets
  10. The bitopological point of view
  11. Strong exactness and zero-dimensionality
  12. The monotopological case
  13. Complete Pervin spaces and complete Frith frames
  14. Stone-type dualities
  15. Stone duality
  16. ...and 9 more sections

Key Result

Proposition 2.1

For every frame $L$ and subset $S\subseteq L$, the frame ${\mathcal{C}}_S L$ has the following universal property: whenever $h:L\rightarrow M$ is a frame map such that $h(s)$ is complemented for all $s\in S$, there is a unique frame homomorphism $\widetilde{h}:{\mathcal{C}}_S L\rightarrow M$ making

Theorems & Definitions (99)

  • Proposition 2.1: Wilson1994TheAT
  • Theorem 2.2
  • Proposition 2.3: borlido21
  • Proposition 2.4: borlido21
  • Theorem 2.5: borlido21
  • Proposition 2.6: borlido21
  • Proposition 2.7: borlido21
  • Proposition 2.8: borlido21
  • Lemma 2.9
  • Lemma 2.10
  • ...and 89 more