Frames and locales: topology without points
(eBook)
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Series:
Published:
Basel : Birkhäuser, [2012].
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eBook
ISBN:
9783034801546, 3034801548, 303480153X, 9783034801539
Content Description:
1 online resource (xix, 398 pages) : illustrations
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Available Online
Description
Until the mid-twentieth century, topological studies were focused on the theory of suitable structures on sets of points. The concept of open set exploited since the 1920s offered an expression of the geometric intuition of a "realistic" place (spot, grain) of non-trivial extent. Imitating the behaviour of open sets and their relations led to a new approach to topology flourishing since the end of the 1950s. It has proved to be beneficial in many respects. Neglecting points, only little information was lost, while deeper insights have been gained; moreover, many results previously dependent on choice principles became constructive. The result is often a smoother, rather than a more entangled, theory. No monograph of this nature has appeared since Johnstone's celebrated Stone Spaces in 1983. The present book is intended as a bridge from that time to the present. Most of the material appears here in book form for the first time or is presented from new points of view. Two appendices provide an introduction to some requisite concepts from order and category theories
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APA Citation (style guide)
Picado, J., & Pultr, A. (2012). Frames and locales: topology without points. Basel, Birkhäuser.
Chicago / Turabian - Author Date Citation (style guide)Picado, Jorge, 1963- and Aleš. Pultr. 2012. Frames and Locales: Topology Without Points. Basel, Birkhäuser.
Chicago / Turabian - Humanities Citation (style guide)Picado, Jorge, 1963- and Aleš. Pultr, Frames and Locales: Topology Without Points. Basel, Birkhäuser, 2012.
MLA Citation (style guide)Picado, Jorge and Aleš Pultr. Frames and Locales: Topology Without Points. Basel, Birkhäuser, 2012.
Note! Citation formats are based on standards as of July 2022. Citations contain only title, author, edition, publisher, and year published. Citations should be used as a guideline and should be double checked for accuracy.
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Language:
English
UPC:
10.1007/978-3-0348-0154-6
Notes
Bibliography
Includes bibliographical references (pages 367-380) and index.
Description
Until the mid-twentieth century, topological studies were focused on the theory of suitable structures on sets of points. The concept of open set exploited since the 1920s offered an expression of the geometric intuition of a "realistic" place (spot, grain) of non-trivial extent. Imitating the behaviour of open sets and their relations led to a new approach to topology flourishing since the end of the 1950s. It has proved to be beneficial in many respects. Neglecting points, only little information was lost, while deeper insights have been gained; moreover, many results previously dependent on choice principles became constructive. The result is often a smoother, rather than a more entangled, theory. No monograph of this nature has appeared since Johnstone's celebrated Stone Spaces in 1983. The present book is intended as a bridge from that time to the present. Most of the material appears here in book form for the first time or is presented from new points of view. Two appendices provide an introduction to some requisite concepts from order and category theories
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English.
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e2dc42dc-508f-9f23-ff10-660d98ffbc0d
Record Information
| Last Sierra Extract Time | Apr 05, 2024 08:52:04 AM |
|---|---|
| Last File Modification Time | Apr 05, 2024 08:59:05 AM |
| Last Grouped Work Modification Time | Apr 05, 2024 08:52:37 AM |
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| 100 | 1 | |a Picado, Jorge,|d 1963-|1 https://id.oclc.org/worldcat/entity/E39PCjxPGktGrjB4D7b7DWTJXd | |
| 245 | 1 | 0 | |a Frames and locales :|b topology without points /|c Jorge Picado, Aleš Pultr. |
| 264 | 1 | |a Basel :|b Birkhäuser,|c [2012]. | |
| 264 | 4 | |c ©2012. | |
| 300 | |a 1 online resource (xix, 398 pages) :|b illustrations | ||
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| 490 | 1 | |a Frontiers in mathematics,|x 1660-8046 | |
| 504 | |a Includes bibliographical references (pages 367-380) and index. | ||
| 505 | 0 | |6 880-01|a I. Spaces and lattices of open sets -- II. Frames and locales, spectra -- III. Sublocals -- IV. Structure of localic morphisms. The categories loc and frm -- V. Separation axioms -- VI. More on sublocales -- VII. Compactness and local compactness -- VIII. (Symmetric) uniformity and nearness -- IX. Paracompactness -- X. More about completion -- XI. Metric frames -- XII. Entourages, asymmetric uniformity -- XIII. Connectedness -- XIV. The frame of reals and real functions -- XV. Localic groups -- Appendices I. Posets -- Appendix II. Categories. | |
| 520 | |a Until the mid-twentieth century, topological studies were focused on the theory of suitable structures on sets of points. The concept of open set exploited since the 1920s offered an expression of the geometric intuition of a "realistic" place (spot, grain) of non-trivial extent. Imitating the behaviour of open sets and their relations led to a new approach to topology flourishing since the end of the 1950s. It has proved to be beneficial in many respects. Neglecting points, only little information was lost, while deeper insights have been gained; moreover, many results previously dependent on choice principles became constructive. The result is often a smoother, rather than a more entangled, theory. No monograph of this nature has appeared since Johnstone's celebrated Stone Spaces in 1983. The present book is intended as a bridge from that time to the present. Most of the material appears here in book form for the first time or is presented from new points of view. Two appendices provide an introduction to some requisite concepts from order and category theories | ||
| 546 | |a English. | ||
| 650 | 0 | |a Topology. | |
| 650 | 0 | |a Mathematics. | |
| 650 | 1 | 2 | |a Mathematics |
| 650 | 6 | |a Topologie. | |
| 650 | 6 | |a Mathématiques. | |
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| 700 | 1 | |a Pultr, Aleš. | |
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| 776 | 0 | 8 | |i Print version:|a Picado, Jorge, 1963-|t Frames and locales.|d Basel : Birkhäuser, ©2012|w (DLC) 2011941088 |
| 830 | 0 | |a Frontiers in mathematics. | |
| 880 | 0 | 0 | |6 505-01/(S|g Machine generated contents note:|g I.|t Spaces and Lattices of Open Sets --|g 1.|t Sober spaces --|g 2.|t axiom TD: another case of spaces easy to reconstruct --|g 3.|t Summing up --|g 4.|t Aside: several technical properties of TD-spaces --|g II.|t Frames and Locales. Spectra --|g 1.|t Frames --|g 2.|t Locales and localic maps --|g 3.|t Points --|g 4.|t Spectra --|g 5.|t unit σ and spatiality --|g 6.|t unit λ and sobriety --|g III.|t Sublocales --|g 1.|t Extremal monomorphisms in Loc --|g 2.|t Sublocales --|g 3.|t co-frame of sublocales --|g 4.|t Images and preimages --|g 5.|t Alternative representations of sublocales --|g 6.|t Open and closed sublocales --|g 7.|t Open and closed localic maps --|g 8.|t Closure --|g 9.|t Preimage as a homomorphism --|g 10.|t Other special sublocales: one-point sublocales, and Boolean ones --|g 11.|t Sublocales as quotients. Factorizing frames is surprisingly easy --|g IV.|t Structure of Localic Morphisms. The Categories Loc and Frm --|g 1.|t Special morphisms. Factorizing in Loc and Frm --|g 2.|t down-set functor and free constructions --|g 3.|t Limits and a colimit in Frm --|g 4.|t Coproducts of frames --|g 5.|t More on the structure of coproduct --|g 6.|t Epimorphisms in Frm --|g V.|t Separation Axioms --|g 1.|t Instead of T1: subfit and fit --|g 2.|t Mimicking the Hausdorff axiom --|g 3.|t I-Hausdorff frames and regular monomorphisms --|g 4.|t Aside: Raney identity --|g 5.|t Quite like the classical case: Regular, completely regular and normal --|g 6.|t categories RegLoc, CRegLoc, HausLoc and FitLoc --|g VI.|t More on Sublocales --|g 1.|t Subspaces and sublocales of spaces --|g 2.|t Spatial and induced sublocales --|g 3.|t Complemented sublocales of spaces are spatial --|g 4.|t zero-dimensionality of Sl(L)op and a few consequences --|g 5.|t Difference and pseudodifference, residua --|g 6.|t Isbell's Development Theorem --|g 7.|t Locales with no non-spatial sublocales --|g 8.|t Spaces with no non-induced sublocales --|g VII.|t Compactness and Local Compactness --|g 1.|t Basics, and a technical lemma --|g 2.|t Compactness and separation --|g 3.|t Kuratowski-Mrowka characterization --|g 4.|t Compactification --|g 5.|t Well below and rather below. Continuous completely regular frames --|g 6.|t Continuous is the same as locally compact. Hofmann-Lawson duality --|g 7.|t One more spatiality theorem --|g 8.|t Supercompactness. Algebraic, superalgebraic and supercontinuous frames --|g VIII.|t (Symmetric) Uniformity and Nearness --|g 1.|t Background --|g 2.|t Uniformity and nearness in the point-free context --|g 3.|t Uniform homomorphisms. Modelling embeddings. Products --|g 4.|t Aside: admitting nearness in a weaker sense --|g 5.|t Compact uniform and nearness frames. Finite covers --|g 6.|t Completeness and completion --|g 7.|t Functoriality. CUniFrm is coreflective in UniFrm --|g 8.|t easy completeness criterion --|g IX.|t Paracompactness --|g 1.|t Full normality --|g 2.|t Paracompactness, and its various guises --|g 3.|t elegant, specifically point-free, characterization of paracompactness --|g 4.|t pleasant surprise: paracompact (co)reflection --|g X.|t More about Completion --|g 1.|t variant of the completion of uniform frames --|g 2.|t Two applications --|g 3.|t Cauchy points and the resulting space --|g 4.|t Cauchy spectrum --|g 5.|t Cauchy completion. The case of countably generated uniformities --|g 6.|t Generalized Cauchy points --|g XI.|t Metric Frames --|g 1.|t Diameters and metric diameters --|g 2.|t Metric spectrum --|g 3.|t Uniform Metrization Theorem --|g 4.|t Metrization theorems for plain frames --|g 5.|t Categories of metric frames --|g XII.|t Entourages. Asymmetric Uniformity --|g 1.|t Entourages --|g 2.|t Uniformities via entourages --|g 3.|t Entourages versus covers --|g 4.|t Asymmetric uniformity: the classical case --|g 5.|t Biframes --|g 6.|t Quasi-uniforniity in the point-free context via paircovers --|g 7.|t adjunction QUnif [→][↔]QUniFrm --|g 8.|t Quasi-uniformity in the point-free context via entourages --|g XIII.|t Connectedness --|g 1.|t few observations about sublocales --|g 2.|t Connected and disconnected locales --|g 3.|t Locally connected locales --|g 4.|t weird example --|g 5.|t few notes --|g XIV.|t Frame of Reals and Real Functions --|g 1.|t frame £(R) of reals --|g 2.|t Properties of £ (R) --|g 3.|t £(R) versus the usual space of reals --|g 4.|t metric uniformity of £(R) --|g 5.|t Continuous real functions --|g 6.|t Cozero elements --|g 7.|t More general real functions --|g 8.|t Notes --|g XV.|t Localic Groups --|g 1.|t Basics --|g 2.|t category of localic groups --|g 3.|t Closed Subgroup Theorem --|g 4.|t multiplication μ is open. The semigroup of open parts --|g 5.|t Uniformities --|g 6.|t Notes --|g Appendix I|t Posets --|g 1.|t Basics --|g 2.|t Zorn's Lemma --|g 3.|t Suprema and infima --|g 4.|t Semilattices, lattices and complete lattices. Completion --|g 5.|t Galois connections (adjunctions) --|g 6.|t (Semi)lattices as algebras. Distributive lattices --|g 7.|t Pseudocomplements and complements. Heyting and Boolean algebras --|g Appendix II|t Categories --|g 1.|t Categories --|g 2.|t Functors and natural transformations --|g 3.|t Some basic constructions --|g 4.|t More special morphisms. Factorization --|g 5.|t Limits and colimits --|g 6.|t Adjunction --|g 7.|t Adjointness and (co)limits --|g 8.|t Reflective and coreflective subcategories --|g 9.|t Monads --|g 10.|t Algebras in a category. |
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