Continuum theory. An introduction.

*(English)*Zbl 0757.54009The book presents a mixture of classical and modern ideas and techniques on continuum theory. Most of the material in the book is in the metric setting. Though many notions are defined for general topological spaces, almost all results are stated and proved for the metric case. Only a minimal knowledge of basic concepts from topology is preassumed. The reader should be familiar with the elementary properties of compactness, connectedness, continuous functions and the topology of metric spaces.

The book is divided into two parts and thirteen chapters. Part One: General Analysis and Part Two: Special Continua and Maps. Chapters I-VI comprise the first part and deal with the general structure of continua. Chapters I–III (I. Examples of continua and nested intersections; II. Inverse limits of continua; III. Decompositions of continua) give general construction techniques; Chapters IV–VI (IV. Limits of sets; V. The boundary bumping theorems: VI. Existence of non-cut points) are concerned with the global analysis of continua; Chapter VII, A general mapping theorem, gives a general method for constructing continuous functions. Chapters VIII-XIII comprise the second part and are concerned with specific types of continua and maps. Chapter VIII–X (VIII. General theory of Peano continua; IX. Graphs; X. Dendrites) are about Peano continua; Chapters XI and XII examine two other important special classes of continua: irreducible continua and arc-like continua; Chapter XIII investigates some special types of maps. The nested intersection technique introduced in Chapter I is used throughout the book to construct continua and maps and to prove theorems. Indecomposable continua and hereditarily indecomposable continua are constructed, investigated and discussed in various places. There are many exercises at the end of each chapter which are an integral part of the book.

Although the book covers only a limited number of topics, the readers will find it an appropriate introduction that gives the uninitiated a foundation on which to build an understanding of continua theory.

The book is divided into two parts and thirteen chapters. Part One: General Analysis and Part Two: Special Continua and Maps. Chapters I-VI comprise the first part and deal with the general structure of continua. Chapters I–III (I. Examples of continua and nested intersections; II. Inverse limits of continua; III. Decompositions of continua) give general construction techniques; Chapters IV–VI (IV. Limits of sets; V. The boundary bumping theorems: VI. Existence of non-cut points) are concerned with the global analysis of continua; Chapter VII, A general mapping theorem, gives a general method for constructing continuous functions. Chapters VIII-XIII comprise the second part and are concerned with specific types of continua and maps. Chapter VIII–X (VIII. General theory of Peano continua; IX. Graphs; X. Dendrites) are about Peano continua; Chapters XI and XII examine two other important special classes of continua: irreducible continua and arc-like continua; Chapter XIII investigates some special types of maps. The nested intersection technique introduced in Chapter I is used throughout the book to construct continua and maps and to prove theorems. Indecomposable continua and hereditarily indecomposable continua are constructed, investigated and discussed in various places. There are many exercises at the end of each chapter which are an integral part of the book.

Although the book covers only a limited number of topics, the readers will find it an appropriate introduction that gives the uninitiated a foundation on which to build an understanding of continua theory.

Reviewer: I.Pop (Iaşi)

##### MSC:

54F15 | Continua and generalizations |

54-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to general topology |