Book:Jorge Picado/Frames and Locales
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Jorge Picado and Aleš Pultr: Frames and Locales
Published $\text {2012}$, Birkhäuser
- ISBN 978-3-0348-0153-9
Subject Matter
Contents
- Preface
- Introduction
- I. Spaces and Lattices of Open Sets
- 1.$\quad$Sober Spaces
- 2.$\quad$The axiom $T_D$: another case of spaces easy to reconstruct
- 3.$\quad$Summing up
- 4.$\quad$Aside: several technical properties of $T_D$-spaces
- II. Frames and Locales. Spectra
- 1.$\quad$Frames
- 2.$\quad$Locales and localic maps
- 3.$\quad$Points
- 4.$\quad$Spectra
- 5.$\quad$The unit $\sigma$ and spatiality
- 6.$\quad$The unit $\lambda$ and sobriety
- III. Sublocales
- 1.$\quad$Extremal monomorphisms in Loc
- 2.$\quad$Sublocales
- 3.$\quad$The co-frame of sublocales
- 4.$\quad$Images and preimages
- 5.$\quad$Alternative representations of sublocales
- 6.$\quad$Open and closed sublocales
- 7.$\quad$Open and closed localic maps
- 8.$\quad$Closure
- 9.$\quad$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.$\quad$Special morphisms. Factorizing in Loc and Frm
- 2.$\quad$The down-set functor and free constructions
- 3.$\quad$Limits and a colimit in Frm
- 4.$\quad$Coproducts of frames
- 5.$\quad$More on the structure of coproduct
- 6.$\quad$Epimorphisms in Frm
- V. Separation Axioms
- 1.$\quad$Instead of $T_1$: subfit and fit
- 2.$\quad$Mimicking the Hausdorff axiom
- 3.$\quad$I-Haudorff frames and regular monomorphisms
- 4.$\quad$Aside: Raney identity
- 5.$\quad$Quite like the classical case: Regular, completely regular and normal
- 6.$\quad$The categories RegLoc,CRegLoc, HausLoc, FitLoc
- VI. More on Sublocales
- 1.$\quad$Subspaces and sublocales of spaces
- 2.$\quad$Spatial and induced sublocales
- 3.$\quad$Complemented sublocales of spaces are spatial
- 4.$\quad$The zero-dimensionlity of $\map {\mathscr {Sl}} L^{op}$ and a few consequences
- 5.$\quad$Difference and pseudodifference, residua
- 6.$\quad$Isbell's Development Theorem
- 7.$\quad$Locales with no non-spatial sublocales
- 8.$\quad$Spaces with no non-induced sublocales
- VII. Compactness and Local Compactness
- 1.$\quad$Basics, and a technical lemma
- 2.$\quad$Compactness and separation
- 3.$\quad$Kuratowski-Mrówka characterization
- 4.$\quad$Compactification
- 5.$\quad$Well below and rather below. Continuous completely regular frames
- 6.$\quad$Continuous is the same as locally compact. Hofmann-Lawson duality
- 7.$\quad$One more spatiality theorem
- 8.$\quad$Supercompactness. Algrbraic, superalgebraic and supercontinuous frames
- VIII. (Symmetric) Uniformity and Nearness
- 1.$\quad$Background
- 2.$\quad$Uniformity and nearness in the point-free context
- 3.$\quad$Uniform homomorphisms. Modelling embeddings. Products
- 4.$\quad$Aside: admitting nearness in a weaker sense
- 5.$\quad$Compact uniform and nearness frames. Finite covers
- 6.$\quad$Completeness and completion
- 7.$\quad$Functoriality. CUniFrm is corefllective in UniFrm
- 8.$\quad$An easy completeness criterion
- IX. Paracompactness
- 1.$\quad$Full normality
- 2.$\quad$Paracompactness, and its various guises
- 3.$\quad$An elegant, specifically point-free, characterization of paracompactness
- 4.$\quad$A pleasant surprise: paracompact (co)reflection
- X. More about Completion
- 1.$\quad$A variant of the completion of uniform frames
- 2.$\quad$Two applications
- 3.$\quad$Cauchy points and the resulting space
- 4.$\quad$Cauchy spectrum
- 5.$\quad$Cauchy completion. The case of countably generated uniformities
- 6.$\quad$Generalized Cauchy points
- XI. Metric Frames
- 1.$\quad$Diameters and metric diameters
- 2.$\quad$Metric spectrum
- 3.$\quad$Uniform Metrization Theorem
- 4.$\quad$Metrization theorems for plain frames
- 5.$\quad$Categories of metric frames
- XII. Entourages. Asymmetric Uniformity
- 1.$\quad$Entourages
- 2.$\quad$Untiformities via entourages
- 3.$\quad$Entourages versus covers
- 4.$\quad$Asymmetric unitformity: the classical case
- 5.$\quad$Biframes
- 6.$\quad$Quasi-uniformity in the point-free context via paircovers
- 7.$\quad$The adjunction QUnif $\rightleftarrows$ QUniFrm
- 8.$\quad$Quasi-uniformity in the point-free context via entourages
- XIII. Connectedness
- 1.$\quad$A few observations about sublocales
- 2.$\quad$Connected and disconnected locales
- 3.$\quad$Locally connected locales
- 4.$\quad$A weird example
- 5.$\quad$A few notes
- XIV. The Frame of Reals and Real Functions
- 1.$\quad$The frame $\map {\mathfrak L} \R$ of reals
- 2.$\quad$Properties of $\map {\mathfrak L} \R$
- 3.$\quad$$\map {\mathfrak L} \R$ versus the usual space of reals
- 4.$\quad$The metric uniformity of $\map {\mathfrak L} \R$
- 5.$\quad$Continuous real functions
- 6.$\quad$Cozero elements
- 7.$\quad$More general real functions
- 8.$\quad$Notes
- XV. Localic Groups
- 1.$\quad$Basics
- 2.$\quad$The category of localic groups
- 3.$\quad$Closed Subgroup Theorem
- 4.$\quad$The multiplication $\mu$ is open. The semigroup of open parts
- 5.$\quad$Uniformities
- 6.$\quad$Notes
- Appendix I. Posets
- 1.$\quad$Basics
- 2.$\quad$Zorn's Lemma
- 3.$\quad$Suprema and infima
- 4.$\quad$Semilatices, lattices and complete lattices. Completion
- 5.$\quad$Galois connections (adjunctions)
- 6.$\quad$(Semi)lattices as algrbras. Distributive lattices
- 7.$\quad$Pseudocomplements and complements. Heyting and Boolean algebras
- Appendix II. Categories
- 1.$\quad$Categories
- 2.$\quad$Functors and natural transformations
- 3.$\quad$Some basic constructions
- 4.$\quad$More special morphisms. Factorization
- 5.$\quad$Limits and colimits
- 6.$\quad$Adjunction
- 7.$\quad$Adjointness and (co)limits
- 8.$\quad$Reflective and coreflective subcategories
- 9.$\quad$Monads
- 10.$\,\,\,$ Algebras in a category
- Bibliography
- List of Symbols
- List of Categories
- Index