nLab
Frames and Locales
Context
Category theory
category theory
Concepts
Universal constructions
Theorems
Extensions
Applications
Topos Theory
This entry provides a hyperlinked index for the book

Jorge Picado, Aleš Pultr,
Frames and Locales. Topology without points,
Frontiers in Mathematics, Birkhäuser (2012)
on basics of locales and pointfree topology.
Contents
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
 Sober spaces?
 The axiom $T_D$: another case of spaces easy to reconstruct
 Summing up
 Aside: several technical properties of $T_D$spaces
II. Frames? and Locales?. Spectra
 Frames?
 Locales? and localic maps?
 Points
 Spectra
 The unit σ and spatiality?
 The unit λ and sobriety?
III. Sublocales?
 Extremal monomorphisms? in Loc
 Sublocales?
 The coframe of sublocales
 Images and preimages
 Alternative representations of sublocales
 Open and closed sublocales
 Open and closed localic maps?
 Closure?
 Preimage as a homomorphism
 Other special sublocales: onepoint sublocales, and Boolean ones
 Sublocales? as quotients. Factorizing frames is surprisingly easy
IV. Structure of Localic Morphisms?. The Categories Loc and Frm
 Special morphisms. Factorizing in Loc and Frm
 The downset functor and free constructions
 Limits? and a colimit in Frm
 Coproducts? of frames
 More on the structure of coproduct
 Epimorphisms in Frm
V. Separation Axioms?
 Instead of $T_1$: subfit? and fit?
 Mimicking the Hausdorff axiom?
 IHausdorff frames and regular monomorphisms
 Aside: Raney identity
 Quite like the classical case: Regular?, completely regular and normal
 The categories RegLoc?, CRegLoc?, HausLoc? and FitLoc?
VI. More on Sublocales?
 Subspaces and sublocales of spaces
 Spatial and induced sublocales
 Complemented sublocales? of spaces are spatial
 The zerodimensionality? of $Sl(L)^op$ and a few consequences
 Diﬀerence and pseudodifference, residua
 Isbell’s Development Theorem?
 Locales with no nonspatial sublocales
 Spaces with no noninduced sublocales
VII. Compactness? and Local Compactness?
 Basics, and a technical lemma
 Compactness? and separation?
 KuratowskiMrówka characterization
 Compactification
 Well below? and rather below?. Continuous completely regular frames
 Continuous is the same as locally compact. HofmannLawson duality?
 One more spatiality theorem
 Supercompactness?. Algebraic, superalgebraic and supercontinuous frames?
VIII. (Symmetric) Uniformity and Nearness
 Background
 Uniformity? and nearness? in the pointfree context
 Uniform homomorphisms. Modelling embeddings. Products
 Aside: admitting nearness in a weaker sense
 Compact uniform and nearness frames. Finite covers
 Completeness? and completion
 Functoriality. CUniFrm? is coreﬂective in UniFrm?
 An easy completeness criterion
IX. Paracompactness?
 Full normality?
 Paracompactness?, and its various guises
 An elegant, specifically pointfree, characterization of paracompactness
 A pleasant surprise: paracompact (co)reflection
X. More about Completion
 A variant of the completion of uniform frames?
 Two applications
 Cauchy points and the resulting space
 Cauchy spectrum?
 Cauchy completion. The case of countably generated uniformities
 Generalized Cauchy points
XI. Metric Frames?
 Diameters and metric diameters
 Metric spectrum
 Uniform Metrization Theorem?
 Metrization theorems? for plain frames
 Categories of metric frames?
XII. Entourages?. Asymmetric Uniformity?
 Entourages?
 Uniformities? via entourages
 Entourages? versus covers
 Asymmetric uniformity?: the classical case
 Biframes?
 Quasiuniformity? in the pointfree context via paircovers?
 The adjunction $QUnif \leftrightarrows QUniFrm$
 Quasiuniformity? in the pointfree context via entourages
XIII. Connectedness?
 A few observations about sublocales
 Connected? and disconnected locales
 Locally connected? locales
 A weird example
 A few notes
XIV. Frame of Reals and Real Functions
 The frame $L(R)$ of reals?
 Properties of $L(R)$
 $L(R)$ versus the usual space of reals
 The metric uniformity of $L(R)$
 Continuous real functions
 Cozero? elements
 More general real functions
 Notes
XV. Localic Groups?
 Basics
 The category of localic groups
 Closed Subgroup Theorem?
 The multiplication μ is open. The semigroup of open parts
 Uniformities
 Notes
Appendix I. Posets
 Basics
 Zorn’s Lemma?
 Suprema? and infima
 Semilattices?, lattices and complete lattices. Completion
 Galois connections (adjunctions)
 (Semi)lattices as algebras. Distributive lattices?
 Pseudocomplements? and complements. Heyting and Boolean algebras
Appendix II. Categories
 Categories
 Functors? and natural transformations
 Some basic constructions
 More special morphisms. Factorization
 Limits? and colimits
 Adjunction?
 Adjointness? and (co)limits
 Reflective and coreflective subcategories
 Monads?
 Algebras in a category