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Hofmann-Mislove through the Lenses of Priestley

G. Bezhanishvili, S. Melzer

TL;DR

The paper reinterprets the Hofmann-Mislove Theorem through Priestley duality by proving that, for any frame L, the poset of Scott-open filters OFilt(L) is isomorphic to the compact saturated subsets of the frame's space of points. It achieves this via a new identification of Scott-open filters with Scott-upsets (special closed upsets) in the Priestley space, and an explicit isomorphism between these upsets and compact saturated sets. This generalizes the Hofmann-Mislove theorem beyond spectral spaces and yields a Zorn-free proof relying only on the Prime Ideal Theorem. The work also clarifies the duality connections between distributive lattices, frames, Esakia spaces, and L-spaces, and provides corollaries about intersections of completely prime filters.

Abstract

We use Priestley duality to give a new proof of the Hofmann-Mislove Theorem.

Hofmann-Mislove through the Lenses of Priestley

TL;DR

The paper reinterprets the Hofmann-Mislove Theorem through Priestley duality by proving that, for any frame L, the poset of Scott-open filters OFilt(L) is isomorphic to the compact saturated subsets of the frame's space of points. It achieves this via a new identification of Scott-open filters with Scott-upsets (special closed upsets) in the Priestley space, and an explicit isomorphism between these upsets and compact saturated sets. This generalizes the Hofmann-Mislove theorem beyond spectral spaces and yields a Zorn-free proof relying only on the Prime Ideal Theorem. The work also clarifies the duality connections between distributive lattices, frames, Esakia spaces, and L-spaces, and provides corollaries about intersections of completely prime filters.

Abstract

We use Priestley duality to give a new proof of the Hofmann-Mislove Theorem.
Paper Structure (5 sections, 13 theorems, 11 equations)

This paper contains 5 sections, 13 theorems, 11 equations.

Table of Contents

  1. Introduction
  2. Priestley duality
  3. Hofmann-Mislove for spectral spaces
  4. Priestley duality for frames
  5. Hofmann-Mislove in full generality

Key Result

Theorem 2.2

$\sf Pries$ is dually equivalent to $\sf Dist$.

Theorems & Definitions (30)

  • Definition 2.1
  • Theorem 2.2: Priestley duality Priestley1970Priestley1972
  • Definition 2.3
  • Theorem 2.4: Cornish Cornish1975
  • Theorem 2.5
  • Definition 3.1
  • Remark 3.2
  • Theorem 3.3: Hofmann-Mislove HofmannMislove1981
  • Definition 4.1
  • Remark 4.2
  • ...and 20 more