Frames and locales
Subtitle: topology without points
By: Jorge Picado, Aleš Pultr
Contributors: Pultr, Aleš
Series: Frontiers in mathematics; Frontiers in mathematics
Publisher: Birkhäuser
Published: 2012
Place of publication: Basel
Language: eng
Number of pages: 398
ISBN-13: 9783034801539
Subjects: Topology
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.
Notes: Includes bibliographical references (p. 367-380) and index.
Table of contents: title: I. Spaces and lattices of open sets; title: II. Frames and locales, spectra; title: III. Sublocals; title: IV. Structure of localic morphisms. The categories loc and frm; title: V. Separation axioms; title: VI. More on sublocales; title: VII. Compactness and local compactness; title: VIII. (Symmetric) uniformity and nearness; title: IX. Paracompactness; title: X. More about completion; title: XI. Metric frames; title: XII. Entourages, asymmetric uniformity; title: XIII. Connectedness; title: XIV. The frame of reals and real functions; title: XV. Localic groups; title: Appendices I. Posets; title: Appendix II. Categories.