Frontiers in Mathematics

Frames and Locales

Topology without points

Authors: Picado, Jorge, Pultr, Ales

  • First monograph in this area in more than two decades
  • Most of the material has not yet been published in book form, particularly the part devoted to the enriched structures
  • Suitable for advanced courses and as a reference text
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eBook $54.99
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  • ISBN 978-3-0348-0154-6
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  • Included format: PDF
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  • Immediate eBook download after purchase
Softcover $74.95
price for USA
  • ISBN 978-3-0348-0153-9
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
About this book

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.

Reviews

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 TD-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)


Table of contents (15 chapters)

  • Spaces and Lattices of Open Sets

    Picado, Jorge (et al.)

    Pages 1-8

  • Frames and Locales. Spectra

    Picado, Jorge (et al.)

    Pages 9-22

  • Sublocales

    Picado, Jorge (et al.)

    Pages 23-48

  • Structure of Localic Morphisms. The Categories Loc and Frm

    Picado, Jorge (et al.)

    Pages 49-72

  • Separation Axioms

    Picado, Jorge (et al.)

    Pages 73-97

Buy this book

eBook $54.99
price for USA (gross)
  • ISBN 978-3-0348-0154-6
  • Digitally watermarked, DRM-free
  • Included format: PDF
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
Softcover $74.95
price for USA
  • ISBN 978-3-0348-0153-9
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
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Bibliographic Information

Bibliographic Information
Book Title
Frames and Locales
Book Subtitle
Topology without points
Authors
Series Title
Frontiers in Mathematics
Copyright
2012
Publisher
Birkhäuser Basel
Copyright Holder
Springer Basel AG
eBook ISBN
978-3-0348-0154-6
DOI
10.1007/978-3-0348-0154-6
Softcover ISBN
978-3-0348-0153-9
Series ISSN
1660-8046
Edition Number
1
Number of Pages
XIX, 398
Topics