nLab Stone Spaces

This page compiles material related to:

• Peter Johnstone:

  Stone Spaces

  Cambridge Studies in Advanced Mathematics 3,

  Cambridge University Press (1982, 1986)

  xxi+370 pages

  ISBN:9780521337793

  MR85f:54002, .

The monograph is ultimately about the Stone representation theorem, but also a standard reference on using locales in place of topological spaces.

(The same author wrote Topos Theory 1977 and the Sketches of an Elephant, both on topos theory).

Although it is a work of mathematics rather than metamathematics, it shows clearly by example how (usually) results about locales do not require the axiom of choice even when analogous results about topological spaces do. Paul Taylor has somewhat imprecisely written of this book

the public theorems about topology […] are marked with an asterisk, although the official meaning of that symbol is a dependence on the axiom of choice. (ASD I, page 3).

Unfortunately for constructive mathematicians, excluded middle is not considered a form of choice by Johnstone.

A trailer for the book (according to its own words) is

• Peter Johnstone, The point of pointless topology , Bull. Amer. Math. Soc. (N.S.) Volume 8, Number 1 (1983), 41-53. (Bulletin AMS, Euclid)

Contents with links to $n$Lab pages

Besides the usual prefaces, bibliography, and indexes, there is a historical introduction, and each chapter concludes with notes on historical and metamathematical aspects. Otherwise, each of 7 chapters is divided into 4 sections, which in turn contain paragraphs that deal with essentially one idea each. For the moment, we list (with minimal processing) the definitions from the index in each section. There will also be some summaries of theorems; as in the book itself, an asterisk here indicates dependence on some form of choice beyond excluded middle (more precisely, a proof that cannot be internalised in an arbitrary boolean topos).

1. Preliminaries

   1. Lattices
      1. poset
      2. join
      3. semilattice
      4. meet, bounded lattice
      5. distributive lattice
      6. complement, Boolean algebra
      7. (none)
      8. symmetric difference
      9. Boolean ring
      10. implication, Heyting algebra
      11. pseudocomplement
      12. (none)
      13. regular element (in a Heyting algebra)
   2. Ideals and filters
      1. ideal (in a lattice or semilattice), lower set, principal ideal
      2. filter (in a lattice or semilattice), prime ideal (in a lattice), prime filter (in a lattice)
      3. * maximal ideal theorem
      4. maximal ideal
      5. * discrete Stone representation theorem
      6. (none)
   3. Some categorical concepts
      1. category, functor, natural transformation
      2. concrete category, locally small category
      3. posets as categories
      4. adjoint functors, free functor, reflective subcategory, equivalence of categories, opposite category
      5. limit, diagram, small category, colimit, regular monomorphism, complete category, finitely complete category
      6. monad, algebra for a monad, monadic adjunction
      7. variety of algebras
      8. algebraic category, equationally presentable category
      9. filtered category, filtered colimit, finitary functor
   4. Free lattices
      1. directed poset, directed join
      2. (none)
      3. suplattice, complete lattice
      4. complete Boolean algebra; free semilattice
      5. free suplattice
      6. free lattice
      7. free complete lattice
      8. free distributive lattice
      9. free boolean algebra
      10. free complete boolean algebra
      11. free Heyting algebra

2. Introduction to locales

   1. Frames and locales
      1. frame, locale, subframe
      2. free frame
      3. point of a locale, completely prime filter, prime element
      4. adjunction between $Loc$ and $Top$
      5. spatial locale
      6. irreducible closed subspace, sober space
      7. soberification, $T_D$-space
      8. specialization order, Alexandroff topology, upper set, upper interval topology
      9. Scott topology
      10. (none)
      11. enrichment of $Loc$ over $Pos$
   2. Sublocales and sites
      1. (none)
      2. nucleus
      3. sublocale
      4. closed nucleus, closed sublocale, open nucleus, open sublocale, dense sublocale, dense nucleus, double-negation nucleus
      5. (none)
      6. (none)
      7. (none)
      8. (none)
      9. (none)
      10. $Loc$ is not well-powered
      11. coverage, site, sheaf
      12. localic product?
      13. (none)
      14. (none)
   3. Coherent locales
      1. compact element (in a lattice)
      2. coherent locale
      3. coherent map? (of locales); local Stone representation theorem for distributive lattices
      4. coherent space, prime spectrum; * spatial Stone representation theorem for distributive lattices
      5. maximal spectrum
      6. normal distributive lattice?
      7. (none)
   4. Stone spaces
      1. totally disconnected space, totally separated space?, zero-dimensional space?
      2. Stone space
      3. (none)
      4. * Stone representation theorem for Boolean algebras, Stone duality
      5. patch topology?
      6. (none)
      7. totally order-separated space?, ordered Stone space?
      8. * $Ord Sto Top \cong Coh Top$
      9. (none)

3. Compact Hausdorff spaces

   1. Compact regular locales
      1. compact locale, regular locale, well inside containment?, zero-dimensional locale?
      2. (none)
      3. strongly Hausdorff locale?
      4. (none)
      5. totally unordered locale?
      6. $Reg Loc$ is complete
      7. $Comp Loc$ is complete (Tychonoff theorem for locales)
      8. localic Stone–Čech compactification
      9. (none)
      10. * $Comp Reg Loc \cong Comp Haus Top$
      11. flat sublocale?
   2. Manes' Theorem?
      1. ultrafilter
      2. filter (on a set), neighbourhood filter, limit (of a filter)
      3. (none)
      4. * $Comp Haus Top$ is monadic
      5. * $Comp Haus Top A$ is monadic for $A$ a variety of algebras
   3. Gleason's Theorem
      1. projective object
      2. (none)
      3. (none)
      4. (none)
      5. extremally disconnected locale?, extremally disconnected space
      6. (none)
      7. projective compact Hausdorff space?
      8. proper map
      9. (none)
      10. (none)
      11. MacNeille cut?, MacNeille completion
   4. Vietoris locales
      1. Vietoris space?
      2. Vietoris topology?, lower interval topology?
      3. Vietoris locale?
      4. (none)
      5. (none)
      6. (none)
      7. (none)
      8. (none)

4. Continuous real-valued functions

   1. Complete regularity and Urysohn's Lemma
      1. (none)
      2. (none)
      3. (none)
      4. scale, really inside containment?
      5. completely regular locale
      6. normal locale
      7. localic Tychonoff embedding theorem?
   2. The Stone–Čech compactification
      1. Stone–?ech compactification?
      2. completely regular ideal?, regular ideal?
      3. completely regular filter?
      4. Wallman base, Wallman compactification
      5. cozero set?
      6. (none)
      7. Alexandroff compactification
      8. (none)
      9. cozero element?, Alexandroff algebra?
      10. (none)
      11. (none)
   3. $C(X)$ and $C^*(X)$
      1. (none)
      2. lattice
      3. Zariski topology
      4. Gelfand–Kolmogorov theorem
      5. fixed maximal ideal?
      6. (none)
      7. real point? (of $\beta X$), realcompact space?, pseudocompact space?
      8. Hewitt realcompactification?
      9. (none)
      10. (none)
      11. (none)
      12. (none)
   4. Gelfand duality
      1. (none)
      2. (none)
      3. Stone–Weierstrass theorem
      4. $C^*$-algebra
      5. (none)
      6. (none)
      7. (none)
      8. (none)
      9. (none)
      10. Stone–Gelfand–Naimark theorem
      11. Dedekind-complete poset?
      12. $\mathit{MI}$-space?

5. Representations of rings

   1. A crash course in sheaf theory
      1. (none)
      2. bundle
      3. trivial bundle, sheaf (on a space), presheaf (on a space)
      4. display space
      5. local homeomorphism
      6. (none)
      7. (none)
      8. direct image functor, inverse image functor
      9. (none)
      10. coherent logic, field
      11. (none)
      12. cartesian logic
      13. regular logic
   2. The Pierce spectrum
      1. (none)
      2. indecomposable ring?
      3. Pierce sheaf?, Pierce representation?, Pierce spectrum
      4. (none)
      5. (none)
      6. von Neumann regular ring
      7. local ring, exchange ring?
      8. neat ideal? (in a ring)
      9. (none)
      10. (none)
   3. The Zariski spectrum
      1. Zariski spectrum?
      2. prime filter (in a ring), radical ideal (in a ring), semiprime ring?
      3. Zariski sheaf
      4. (none)
      5. Zariski representation?, local homomorphism? (of rings)
      6. nilradical
      7. Gelfand ring?
      8. (none)
      9. (none)
      10. (none)
      11. integral domain, domain spectrum?, domain representable ring?
      12. field spectrum?
      13. (none)
   4. Ordered rings and real rings
      1. ordered ring, positive cone
      2. concave prime filter? (in a ring), Brumfiel spectrum?
      3. shadow, local ordered ring?
      4. $L$-ring?
      5. (none)
      6. $L$-ideal? (in a ring), $F$-ring?, Keimel spectrum?, irreducible? $L$-ideal
      7. (none)
      8. (none)
      9. $L$-local? $F$-ring
      10. $L$-simple? $F$-ring
      11. formally real field, real-closed field, real point? (of $spec A$), strictly positive filter? (in a ring)
      12. real spectrum?
      13. formally real local ring?, ordered local ring

6. Profiniteness and duality

   1. Ind-objects and pro-objects
      1. (none)
      2. ind-object
      3. finitely continuous functor
      4. (none)
      5. final functor
      6. (none)
      7. (none)
      8. cocompletion, finitely-presentable object
      9. pro-object
   2. Profinite sets and algebras
      1. (none)
      2. (none)
      3. profinite set
      4. (none)
      5. Jónsson–Tarski algebra
      6. congruence
      7. (none)
      8. (none)
      9. (none)
      10. (none)
   3. Stone-type dualities
      1. (none)
      2. (none)
      3. (none)
      4. (none)
      5. (none)
      6. algebraic lattice
      7. (none)
   4. General concrete dualities
      1. coseparator, schizophrenic object
      2. (none)
      3. (none)
      4. (none)
      5. idempotent adjunction
      6. Sierpi?ski topology, Sierpi?ski space
      7. (none)
      8. (none)
      9. (none)
      10. (none)
      11. (none)

7. Continuous lattices

   1. Compact topological (semi)lattices?
      1. ordered space?, topological poset?, order-Hasudorff space?
      2. order-normal space?
      3. (none)
      4. (none)
      5. continuously distributive lattice?
      6. (none)
      7. (none)
      8. (none)
      9. (none)
      10. completely distributive lattice
      11. interval topology?
      12. (none)
      13. (none)
      14. (none)
      15. (none)
      16. (none)
      17. (none)
   2. Continuous posets and lattices
      1. ideal (in a poset)
      2. way below containment, continuous poset, continuous lattice
      3. algebraic poset?
      4. (none)
      5. filter (in a poset), Scott-open filter?
      6. (none)
      7. (none)
      8. (none)
      9. (none)
      10. Lawson map?
      11. continuous semilattice?
      12. stably continuous poset?
   3. Lawson semilattices
      1. (none)
      2. (none)
      3. Lawson topology?, Lawson semilattice?
      4. (none)
      5. (none)
      6. (none)
      7. (none)
      8. (none)
   4. Locally compact locales
      1. (none)
      2. locally compact locale
      3. (none)
      4. (none)
      5. (none)
      6. stably locally compact locale?
      7. injective sober space?
      8. (none)
      9. injective locale?
      10. exponentiable object, exponentiable locale?
      11. (none)
      12. exponentiable space

category: reference