© 2012

Frames and Locales

Topology without points


  • 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


Part of the Frontiers in Mathematics book series (FM)

Table of contents

  1. Front Matter
    Pages i-xix
  2. Jorge Picado, Aleš Pultr
    Pages 1-8
  3. Jorge Picado, Aleš Pultr
    Pages 9-22
  4. Jorge Picado, Aleš Pultr
    Pages 23-48
  5. Jorge Picado, Aleš Pultr
    Pages 49-72
  6. Jorge Picado, Aleš Pultr
    Pages 73-97
  7. Jorge Picado, Aleš Pultr
    Pages 99-123
  8. Jorge Picado, Aleš Pultr
    Pages 125-144
  9. Jorge Picado, Aleš Pultr
    Pages 145-167
  10. Jorge Picado, Aleš Pultr
    Pages 169-181
  11. Jorge Picado, Aleš Pultr
    Pages 183-202
  12. Jorge Picado, Aleš Pultr
    Pages 203-226
  13. Jorge Picado, Aleš Pultr
    Pages 227-252
  14. Jorge Picado, Aleš Pultr
    Pages 253-267
  15. Jorge Picado, Aleš Pultr
    Pages 269-296
  16. Jorge Picado, Aleš Pultr
    Pages 297-314
  17. Back Matter
    Pages 315-398

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 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.


Categorical topology Lattice theory Order theory

Authors and affiliations

  1. 1., CMUC, Department of MathematicsUniversity of CoimbraCoimbraPortugal
  2. 2., Department of Applied Mathematics (KAM)MFF Karlovy UniversityPrague 1Czech Republic

Bibliographic information

  • Book Title Frames and Locales
  • Book Subtitle Topology without points
  • Authors Jorge Picado
    Aleš Pultr
  • Series Title Frontiers in Mathematics
  • DOI
  • Copyright Information Springer Basel AG 2012
  • Publisher Name Springer, Basel
  • eBook Packages Mathematics and Statistics Mathematics and Statistics (R0)
  • Softcover ISBN 978-3-0348-0153-9
  • eBook ISBN 978-3-0348-0154-6
  • Series ISSN 1660-8046
  • Series E-ISSN 1660-8054
  • Edition Number 1
  • Number of Pages XIX, 398
  • Number of Illustrations 0 b/w illustrations, 0 illustrations in colour
  • Topics Topology
  • Buy this book on publisher's site


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)