Frames and Locales - Topology without points

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 twenties 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 fifties.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.

List of contents

Preface.- Introduction.- I. Spaces and lattices of open sets.- II. Frames and locales. Spectra.- III. Sublocales.- 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, non-symmetric uniformity.- XIII. Connectedness.- XIV. The frame of reals and real functions.- XV. Localic groups.- Appendix I: Posets.- Appendix II: Categories.- Bibliography.- Index of Notation.- Index.

Summary

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 twenties 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 fifties.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.

Additional text

From the reviews:
“The book covers, in a comprehensive and self-contained way, the development of point-free topology in the last 30 years … . The book ends with two short appendices containing those facts about partially ordered sets and categories needed in the book. In conclusion, this is a very good book; it is nicely written and is highly recommended for anyone wishing to gain an overview of point-free topology.” (Javier Gutiérrez García, Mathematical Reviews, Issue 2012 j)
“The book starts with a recount of sobriety and the T_D-axiom, which is then quickly followed by the definition of the category of locales, various facets of spectra, the usual adjunctions and several criteria for spatiality. … this book is an erudite account of the current status of pointfree topology, written beautifully in the authors’ inimitable style. It is both easily accessible to a complete beginner, and an excellent source of reference for the mature pointfree practitioner.” (Themba Dube, Zentralblatt MATH, Vol. 1231, 2012)
“In a total of fifteen chapters the reader is taken from spaces and lattices of open sets, through a thorough discussion of the notions of frames and locales … . book ends (modulo its appendices) with a discussion of localic groups … together with frames, that’s what this entire book is ultimately all about. … is quite-well written: it’s a clear and thorough treatment of what is really rather an important subject, which is to say, a greatly underappreciated one.” (Michael Berg, The Mathematical Association of America, April, 2012)

Report

From the reviews:
"The book covers, in a comprehensive and self-contained way, the development of point-free topology in the last 30 years ... . The book ends with two short appendices containing those facts about partially ordered sets and categories needed in the book. In conclusion, this is a very good book; it is nicely written and is highly recommended for anyone wishing to gain an overview of point-free topology." (Javier Gutiérrez García, Mathematical Reviews, Issue 2012 j)
"The book starts with a recount of sobriety and the T_D-axiom, which is then quickly followed by the definition of the category of locales, various facets of spectra, the usual adjunctions and several criteria for spatiality. ... this book is an erudite account of the current status of pointfree topology, written beautifully in the authors' inimitable style. It is both easily accessible to a complete beginner, and an excellent source of reference for the mature pointfree practitioner." (Themba Dube, Zentralblatt MATH, Vol. 1231, 2012)
"In a total of fifteen chapters the reader is taken from spaces and lattices of open sets, through a thorough discussion of the notions of frames and locales ... . book ends (modulo its appendices) with a discussion of localic groups ... together with frames, that's what this entire book is ultimately all about. ... is quite-well written: it's a clear and thorough treatment of what is really rather an important subject, which is to say, a greatly underappreciated one." (Michael Berg, The Mathematical Association of America, April, 2012)

Product details

• Authors: Jorg Picado, Jorge Picado, Ales Pultr, Aleš Pultr
• Publisher: Springer, Basel
• Languages: English
• Product format: Paperback / Softback
• Released: 01.12.2011
• EAN: 9783034801539
• ISBN: 978-3-0-3480153-9
• No. of pages: 398
• Dimensions: 169 mm x 240 mm x 24 mm
• Weight: 692 g
• Illustrations: XIX, 398 p.
• Series: Frontiers in Mathematics; Frontiers in Mathematics
• Subjects: Natural sciences, medicine, IT, technology > Mathematics > Geometry; B, Mathematics and Statistics, Topology, Categorical topology;Lattice theory;Order theory