A pointfree theory of Pervin spaces

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

“…Symmetrization is for Pervin spaces and Frith frames, what uniformization is for quasiuniform spaces and quasi-uniform frames, respectively. This has been considered in [19] for Pervin spaces and in [7] for Frith frames. In particular, it has been shown that the categories of symmetric Pervin spaces and of symmetric Frith frames are equivalent to the categories of transitive and totally bounded uniform spaces and frames, respectively.…”

“…A filter F ⊆ P(X ) is a Cauchy filter if it is proper and, for every S ∈ S, either S or its complement is in F. We say that a Cauchy filter F converges to the point x ∈ X if every open neighborhood U ∈ S (X ) of x belongs to F. Finally, a Pervin space (X , S) is said to be Cauchy complete if every Cauchy filter converges, and a Cauchy completion of (X , S) is a dense extremal monomorphism c : (X , S) → (Y , T ) into a Cauchy complete Pervin space (Y , T ). 7 In the following, we refer to the symmetrization of a Pervin space, defined in Sect. 2.5.…”

“…the category of Frith frames in that of transitive and totally bounded quasi-uniform frames, which is a coreflection but not an equivalence. It is also shown in [7] that the classical dual adjunction between topological spaces and frames naturally extends to a dual adjunction between Pervin spaces and Frith frames. In fact, this is what justifies calling Frith frames the pointfree version of Pervin spaces.…”

“…As representatives of quasi-uniform structures, Pervin spaces and Frith frames also inherit natural notions of completeness. It is observed in [19] that T 0 and complete Pervin spaces can be identified with spectral spaces, while in [7] it is shown that complete Frith frames can be identified with bounded distributive lattices. In particular, thanks to Stone duality for bounded distributive lattices, it follows that the categories of T 0 and complete Pervin spaces and of complete Frith frames are dual to each other.…”

“…This reduces the amount of background required from the reader, leav-ing the details of existing connections for those already familiar with quasi-uniformities. For a detailed study of Pervin spaces, Frith frames, and corresponding quasi-uniform structures, we refer to [7,11,19].…”