# nLab Stone Spaces

Contents with links to Lab pages
• Peter Johnstone, Stone Spaces, Cambridge Studies in Advanced Mathematics 3, Cambridge University Press 1982. xxi+370 pp. MR85f:54002, reprinted 1986.

(The same author wrote Topos theory 1977 and the Elephant).

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

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

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
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
4. Free lattices
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)
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?
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)
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

Last revised on May 21, 2017 at 06:35:48. See the history of this page for a list of all contributions to it.