Exact and Strongly Exact Filters

Moshier, M. Andrew; Pultr, Aleš; Suarez, Anna Laura

doi:10.1007/s10485-020-09602-0

“…Recall that a filter F ⊆ L on a frame L is Scott-open if whenever D ⊆ F is a directed subset whose join belongs to F , the intersection D ∩ F is nonempty. Proposition 5.12 ( [13,14]). Scott-open filters are closed under strongly exact meets.…”

Section: It Only Remains To Show the Inclusion Skmentioning

confidence: 99%

“…Proof. It is shown in [13,Lemma 3.4] that every Scott-open filter F ⊆ L is of the form {x ∈ L | ∆ x ⊆ a∈P ∆ a }, for some subset P ⊆ L. In turn, filters of this form are shown in [14,Theorem 4.5] to be closed under strongly exact meets. Remark 5.13.…”

Section: It Only Remains To Show the Inclusion Skmentioning

confidence: 99%

A pointfree theory of Pervin spaces

Borlido

¹

,

Suarez

²

2023

Quaestiones Mathematicae

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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.

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“…Proof It is shown in [14,Lemma 3.4] that every Scott-open filter F ⊆ L is of the form {x ∈ L | x ⊆ a∈P a }, for some subset P ⊆ L. In turn, filters of this form are shown in [15,Theorem 4.5] to be closed under strongly exact meets. Remark 5.…”

Section: Lemma 514 Complete Frith Frames Are Strongly Exactmentioning

confidence: 99%

Pervin Spaces and Frith Frames: Bitopological Aspects and Completion

Borlido,

Suarez

2023

Appl Categor Struct

0

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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.

show abstract