Separation in Point-Free Topology

ISBN-13: 9783030534813 / Angielski / Miękka / 2022 / 304 str.

Jorge Picado; Ales Pultr

Recenzje

"The book is written in a very engaging style. ... the book is appealing to experts who will find a clear exposition of numerous results elegantly bundled into a coherent story. The book is also valuable to those immigrating from point-centric topology to the point-free realm." (Ittay Weiss, MAA Reviews, February 19, 2023)
"As the title promises, this book deals with a thorough study of different types of separation axioms in pointfree topology ... . The authors do an extremely good job collecting results scattered throughout the literature and presenting them here in a uniform way ... drawing attention to the parallelisms and differences between the classical and pointfree worlds." (Mark Sioen, Mathematical Reviews, October, 2022)
"The book is very well-written with great attention to detail, making it a pleasure to read and, on top of covering a vast body of material ... . the monograph under review ... can only be highly recommended: they constitute a comprehensive source of information and insights regarding pointfree topology, which will be of great interest and value to researchers already working in the field, to mathematicians who want to study pointfree topology and to general topologists ... ." (Mark Sioen, zbMATH 1486.54001, 2022)

Spis treści

Introduction.- Separation in spaces.- Subfitness, and basics of fitness.- Subfitness, and basics of fitness.- Summarizing low separation.- Regularity and fitness.- Complete regularity.- Complete regularity.- Normality.- Complete regularity.- Scatteredness. Joins of closed sublocales.- Subfit, fit, open and complete.- Appendix.

Blurb

This book is the first systematic treatment of this area so far scattered in a vast number of articles. As in classical topology, concrete problems require restricting the (generalized point-free) spaces by various conditions playing the roles of classical separation axioms. These are typically formulated in the language of points; but in the point-free context one has either suitable translations, parallels, or satisfactory replacements. The interrelations of separation type conditions, their merits, advantages and disadvantages, and consequences are discussed.

Highlights of the book include a treatment of the merits and consequences of subfitness, various approaches to the Hausdorff's axiom, and normality type axioms. Global treatment of the separation conditions put them in a new perspective, and, a.o., gave some of them unexpected importance. The text contains a lot of quite recent results; the reader will see the directions the area is taking, and may find inspiration for her/his further work.

The book will be of use for researchers already active in the area, but also for those interested in this growing field (sometimes even penetrating into some parts of theoretical computer science), for graduate and PhD students, and others. For the reader's convenience, the text is supplemented with an Appendix containing necessary background on posets, frames and locales.