nLab
Stone Spaces
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
In [Joh82] 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.
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 ).
Preliminaries
Lattices
poset
join
semilattice
meet , bounded lattice
distributive lattice
complement , Boolean algebra
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symmetric difference
Boolean ring
implication , Heyting algebra
pseudocomplement
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regular element (in a Heyting algebra)
Ideals and filters
ideal (in a lattice or semilattice), lower set , principal ideal
filter (in a lattice or semilattice), prime ideal (in a lattice), prime filter (in a lattice)
* maximal ideal theorem
maximal ideal
* discrete Stone representation theorem
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Some categorical concepts
category , functor , natural transformation
concrete category , locally small category
posets as categories
adjoint functors , free functor , reflective subcategory , equivalence of categories , opposite category
limit , diagram , small category , colimit , regular monomorphism , complete category , finitely complete category
monad , algebra for a monad , monadic adjunction
variety of algebras
algebraic category , equationally presentable category
filtered category , filtered colimit , finitary functor
Free lattices
directed poset , directed join
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suplattice , complete lattice
complete Boolean algebra ; free semilattice
free suplattice
free lattice
free complete lattice
free distributive lattice
free boolean algebra
free complete boolean algebra
free Heyting algebra
Introduction to locales
Frames and locales
frame , locale , subframe
free frame
point of a locale , completely prime filter , prime element
adjunction between $Loc$ and $Top$
spatial locale
irreducible closed subspace , sober space
soberification , $T_D$ -space
specialization order , Alexandroff topology , upper set , upper interval topology
Scott topology
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enrichment of $Loc$ over $Pos$
Sublocales and sites
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nucleus
sublocale
closed nucleus , closed sublocale , open nucleus , open sublocale , dense sublocale , dense nucleus , double-negation nucleus
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$Loc$ is not well-powered
coverage , site , sheaf
localic product?
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Coherent locales
compact element (in a lattice)
coherent locale?
coherent map? (of locales); local Stone representation theorem for distributive lattices
coherent space , prime spectrum ; * spatial Stone representation theorem for distributive lattices
maximal spectrum
normal distributive lattice?
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Stone spaces
totally disconnected space , totally separated space? , zero-dimensional space?
Stone space
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* Stone representation theorem for Boolean algebras, Stone duality
patch topology?
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totally order-separated space? , ordered Stone space?
* $Ord Sto Top \cong Coh Top$
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Compact Hausdorff spaces
Compact regular locales
compact locale , regular locale , well inside containment? , zero-dimensional locale?
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strongly Hausdorff locale?
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totally unordered locale?
$Reg Loc$ is complete
$Comp Loc$ is complete (Tychonoff theorem for locales)
localic Stone–Čech compactification
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* $Comp Reg Loc \cong Comp Haus Top$
flat sublocale?
Manes' Theorem?
ultrafilter
filter (on a set), neighbourhood filter , limit (of a filter)
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* $Comp Haus Top$ is monadic
* $Comp Haus Top A$ is monadic for $A$ a variety of algebras
Gleason's Theorem
projective object
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extremally disconnected locale? , extremally disconnected space
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projective compact Hausdorff space?
proper map
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MacNeille cut? , MacNeille completion
Vietoris locales
Vietoris space?
Vietoris topology? , lower interval topology?
Vietoris locale?
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Continuous real-valued functions
Complete regularity and Urysohn's Lemma
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scale? , really inside containment?
completely regular locale
normal locale
localic Tychonoff embedding theorem?
The Stone–Čech compactification
Stone–Čech compactification
completely regular ideal? , regular ideal?
completely regular filter?
Wallman base , Wallman compactification
cozero set?
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Alexandroff compactification
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cozero element? , Alexandroff algebra?
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$C(X)$ and $C^*(X)$
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lattice
Zariski topology
Gelfand–Kolmogorov theorem?
fixed maximal ideal?
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real point? (of $\beta X$ ), realcompact space? , pseudocompact space?
Hewitt realcompactification?
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Gelfand duality
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Stone–Weierstrass theorem
$C^*$ -algebra
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Stone–Gelfand–Naimark theorem
Dedekind-complete poset?
$\mathit{MI}$ -space?
Representations of rings
A crash course in sheaf theory
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bundle
trivial bundle , sheaf (on a space), presheaf (on a space)
display space
local homeomorphism
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direct image functor , inverse image functor
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coherent logic , field
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cartesian logic
regular logic
The Pierce spectrum
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indecomposable ring?
Pierce sheaf? , Pierce representation? , Pierce spectrum
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von Neumann regular ring?
local ring , exchange ring?
neat ideal? (in a ring)
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The Zariski spectrum
Zariski spectrum
prime filter (in a ring), radical ideal? (in a ring), semiprime ring?
Zariski sheaf?
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Zariski representation? , local homomorphism? (of rings)
nilradical
Gelfand ring?
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integral domain , domain spectrum? , domain representable ring?
field spectrum?
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Ordered rings and real rings
ordered ring? , positive cone
concave prime filter? (in a ring), Brumfiel spectrum?
shadow? , local ordered ring?
$L$ -ring?
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$L$ -ideal? (in a ring), $F$ -ring? , Keimel spectrum? , irreducible? $L$ -ideal
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$L$ -local? $F$ -ring
$L$ -simple? $F$ -ring
formally real field , real-closed field , real point? (of $spec A$ ), strictly positive filter? (in a ring)
real spectrum?
formally real local ring? , ordered local ring?
Profiniteness and duality
Ind-objects and pro-objects
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ind-object
finitely continuous functor?
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final functor
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cocompletion? , finitely-presentable object
pro-object
Profinite sets and algebras
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profinite set
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Jónsson–Tarski algebra
congruence
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Stone-type dualities
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algebraic lattice
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General concrete dualities?
coseparator , schizophrenic object
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idempotent adjunction
Sierpiński topology , Sierpiński space
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Continuous lattices
Compact topological (semi)lattices?
ordered space? , topological poset? , order-Hasudorff space?
order-normal space?
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continuously distributive lattice?
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completely distributive lattice
interval topology?
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Continuous posets and lattices
ideal (in a poset)
way below containment , continuous poset , continuous lattice
algebraic poset?
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filter (in a poset), Scott-open filter?
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Lawson map?
continuous semilattice?
stably continuous poset?
Lawson semilattices
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Lawson topology? , Lawson semilattice?
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Locally compact locales
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locally compact locale
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stably locally compact locale?
injective sober space?
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injective locale?
exponentiable object , exponentiable locale?
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exponentiable space
Revised on September 17, 2011 08:32:29
by
Toby Bartels
(71.31.209.1)