Topics on Continua: 1st Edition (Hardback) book cover

Topics on Continua

1st Edition

By Sergio Macias

Chapman and Hall/CRC

376 pages

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Hardback: 9780849337383
pub: 2005-05-26
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Description

Specialized as it might be, continuum theory is one of the most intriguing areas in mathematics. However, despite being popular journal fare, few books have thoroughly explored this interesting aspect of topology.

In Topics on Continua, Sergio Macías, one of the field’s leading scholars, presents four of his favorite continuum topics: inverse limits, Jones’s set function T, homogenous continua, and n-fold hyperspaces, and in doing so, presents the most complete set of theorems and proofs ever contained in a single topology volume. Many of the results presented have previously appeared only in research papers, and some appear here for the first time.

After building the requisite background and exploring the inverse limits of continua, the discussions focus on Professor Jones's set function T and continua for which T is continuous. An introduction to topological groups and group actions lead to a proof of Effros's Theorem, followed by a presentation of two decomposition theorems. The author then offers an in-depth study of n-fold hyperspaces. This includes their general properties, conditions that allow points of n-fold symmetric products to be arcwise accessible from their complement, points that arcwise disconnect the n-fold hyperspaces, the n-fold hyperspaces of graphs, and theorems relating n-fold hyperspaces and cones. The concluding chapter presents a series of open questions on each topic discussed in the book.

With more than a decade of teaching experience, Macías is able to put forth exceptionally cogent discussions that not only give beginning mathematicians a strong grounding in continuum theory, but also form an authoritative, single-source guide through some of topology's most captivating facets.

Reviews

“The book consists of four expository papers … topics: inverse limits, Jones’ set function T, homogeneous continua and n-fold hyperspaces. … It includes a quick survey of some topics on metric spaces and general topology. … It presents standard results and shows that the operation of taking inverse limits commutes with the operation of taking finite products, cones and hyperspaces. … problems to the topics of the book. …”

— Alejandro Illanes, Mexico, in Zentralblatt MATH, Vol. 1081, 2006/07

Table of Contents

PRELIMINARIES

Product Topology

Continuous Decompositions

Homotopy and Fundamental Group

Geometric Complexes and Polyhedra

Complete Metric Spaces

Compacta

Continua

Hyperspaces

References

INVERSE LIMITS AND RELATED TOPICS

Inverse Limits

Inverse Limits and the Cantor Set

Inverse Limits and Other Operations

Chainable Continua

Circularly Chainable and P–like Continua

Universal and A–H Essential Maps

References

JONES’S SET FUNCTION T

The Set Function T

Continuity of T

Applications

References

A THEOREM OF E. G. EFFROS

Topological Groups

Group Actions and a Theorem of Effros

References

DECOMPOSITION THEOREMS

Jones’s Theorem

Detour to Covering Spaces

Rogers’s Theorem

Case and Minc–Rogers Continua

Covering Spaces of Some Homogeneous Continua

References

n–FOLD HYPERSPACES

General Properties

Unicoherence

Aposyndesis

Arcwise Accessibility

Points that Arcwise Disconnect

C*n–smoothness

Retractions

Graphs

Cones

References

QUESTIONS

Inverse Limits

The Set Function T

Homogeneous Continua

n–fold Hyperspaces

References

Index

About the Series

Chapman & Hall/CRC Pure and Applied Mathematics

Learn more…

Subject Categories

BISAC Subject Codes/Headings:
MAT012000
MATHEMATICS / Geometry / General
MAT037000
MATHEMATICS / Functional Analysis