nLab Frames and Locales

This entry is to record the reference:

• Jorge Picado, Aleš Pultr:

  
  

  Frames and Locales: Topology without points

  
  

  Frontiers in Mathematics

  Birkhäuser (2012)

  doi:10.1007/978-3-0348-0154-6

  slides: pdf

on frames, locales and their use in pointfree topology.

The following lists chapterwise linked lists of keywords to relevant and related existing entries, as far as they already exist.

I. Spaces and Lattices of Open Sets

1. Sober spaces
2. The axiom $T_D$: another case of spaces easy to reconstruct
3. Summing up
4. Aside: several technical properties of $T_D$-spaces

II. Frames and Locales. Spectra

1. Frames
2. Locales and localic maps
3. Points
4. Spectra
5. The unit σ and spatiality
6. The unit λ and sobriety

III. Sublocales

1. Extremal monomorphisms in Loc
2. Sublocales
3. The co-frame of sublocales
4. Images and preimages
5. Alternative representations of sublocales
6. Open and closed sublocales
7. Open and closed localic maps
8. Closure
9. Preimage as a homomorphism
10. Other special sublocales: one-point sublocales, and Boolean ones
11. Sublocales as quotients. Factorizing frames is surprisingly easy

IV. Structure of Localic Morphisms. The Categories Loc and Frm

1. Special morphisms. Factorizing in Loc and Frm
2. The down-set functor and free constructions
3. Limits and a colimit in Frm
4. Coproducts of frames
5. More on the structure of coproduct
6. Epimorphisms in Frm

V. Separation Axioms

1. Instead of $T_1$: subfit and fit
2. Mimicking the Hausdorff axiom
3. I-Hausdorff frames and regular monomorphisms
4. Aside: Raney identity
5. Quite like the classical case: Regular, completely regular and normal
6. The categories RegLoc, CRegLoc, HausLoc and FitLoc

VI. More on Sublocales

1. Subspaces and sublocales of spaces
2. Spatial and induced sublocales
3. Complemented sublocales of spaces are spatial
4. The zero-dimensionality of $Sl(L)^op$ and a few consequences
5. Difference and pseudodifference, residua
6. Isbell’s Development Theorem
7. Locales with no non-spatial sublocales
8. Spaces with no non-induced sublocales

VII. Compactness and Local Compactness

1. Basics, and a technical lemma
2. Compactness and separation
3. Kuratowski-Mrówka characterization
4. Compactification
5. Well below and rather below. Continuous completely regular frames
6. Continuous is the same as locally compact. Hofmann-Lawson duality
7. One more spatiality theorem
8. Supercompactness. Algebraic, superalgebraic and supercontinuous frames

VIII. (Symmetric) Uniformity and Nearness

1. Background
2. Uniformity and nearness in the point-free context
3. Uniform homomorphisms. Modelling embeddings. Products
4. Aside: admitting nearness in a weaker sense
5. Compact uniform and nearness frames. Finite covers
6. Completeness and completion
7. Functoriality. CUniFrm is coreflective in UniFrm
8. An easy completeness criterion

IX. Paracompactness

1. Full normality
2. Paracompactness, and its various guises
3. An elegant, specifically point-free, characterization of paracompactness
4. A pleasant surprise: paracompact (co)reflection

X. More about Completion

1. A variant of the completion of uniform frames
2. Two applications
3. Cauchy points and the resulting space
4. Cauchy spectrum
5. Cauchy completion. The case of countably generated uniformities
6. Generalized Cauchy points

XI. Metric Frames

1. Diameters and metric diameters
2. Metric spectrum
3. Uniform Metrization Theorem
4. Metrization theorems for plain frames
5. Categories of metric frames

XII. Entourages. Asymmetric Uniformity

1. Entourages
2. Uniformities via entourages
3. Entourages versus covers
4. Asymmetric uniformity: the classical case
5. Biframes
6. Quasi-uniformity in the point-free context via paircovers
7. The adjunction $QUnif \leftrightarrows QUniFrm$
8. Quasi-uniformity in the point-free context via entourages

XIII. Connectedness

1. A few observations about sublocales
2. Connected and disconnected locales
3. Locally connected locales
4. A weird example
5. A few notes

XIV. Frame of Reals and Real Functions

1. The frame $L(R)$ of reals
2. Properties of $L(R)$
3. $L(R)$ versus the usual space of reals
4. The metric uniformity of $L(R)$
5. Continuous real functions
6. Cozero elements
7. More general real functions
8. Notes

XV. Localic Groups

1. Basics
2. The category of localic groups
3. Closed Subgroup Theorem
4. The multiplication μ is open. The semigroup of open parts
5. Uniformities
6. Notes

Appendix I. Posets

1. Basics
2. Zorn’s Lemma
3. Suprema and infima
4. Semilattices, lattices and complete lattices. Completion
5. Galois connections (adjunctions)
6. (Semi)lattices as algebras. Distributive lattices
7. Pseudocomplements and complements. Heyting and Boolean algebras

Appendix II. Categories

1. Categories
2. Functors and natural transformations
3. Some basic constructions
4. More special morphisms. Factorization
5. Limits and colimits
6. Adjunction
7. Adjointness and (co)limits
8. Reflective and coreflective subcategories
9. Monads
10. Algebras in a category

category: reference