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From Geometry to Topology

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From Geometry to Topology Cover

 

Synopses & Reviews

Publisher Comments:

An essential element of advanced studies in mathematics, topology tends to receive a highly formal and abstract treatment, discouraging students from grasping even the simpler ideas or getting any real "feel" for the subject. This volume, on the other hand, offers students a bridge from the familiar concepts of geometry to the formalized study of topology. It begins by exploring simple transformations of familiar figures in ordinary Euclidean space and develops the idea of congruence classes. By gradually expanding the number of "permitted" transformations, these classes increase and their relationships to topological properties develop in an intuitive manner. Imaginative introductions to selected topological subjects complete the intuitive approach, and students then advance to a more conventional presentation. An invaluable initiation into the formal study of topology for prospective and first-year mathematics students.

Synopsis:

Introductory text for first-year math students uses intuitive approach, bridges the gap from familiar concepts of geometry to topology. Exercises and Problems. Includes 101 black-and-white illustrations. 1974 edition.

Synopsis:

This excellent introduction to topology eases first-year math students and general readers into the subject by surveying its concepts in a descriptive and intuitive way, attempting to build a bridge from the familiar concepts of geometry to the formalized study of topology. The first three chapters focus on congruence classes defined by transformations in real Euclidean space. As the number of permitted transformations increases, these classes become larger, and their common topological properties become intuitively clear. Chapters 4–12 give a largely intuitive presentation of selected topics. In the remaining five chapters, the author moves to a more conventional presentation of continuity, sets, functions, metric spaces, and topological spaces. Exercises and Problems. 101 black-and-white illustrations. 1974 edition.

Synopsis:

This introduction to topology eases readers into the subject by building a bridge from the familiar concepts of geometry to the formalized study of topology. Focuses on congruence classes defined by transformations in real Euclidean space, continuity, sets, functions, metric spaces, and topological spaces, and more. Exercises and Problems. Includes 101 black-and-white illustrations. 1974 edition.

Table of Contents

  Author's Preface

  Acknowledgements

1 Congruence Classes

    What geometry is about

    Congruence

    "The rigid transformations: translation, reflection, rotation"

    Invariant properties

    Congruence as an equivalence relation

    Congruence classes as the concern of Euclidean geometry

2 Non-Euclidean Geometries

    Orientation as a property

    Orientation geometry divides congruence classes

    Magnification (and contraction) combine congruence classes

    Invariants of similarity geometry

    Affine and projective transformations and invariants

    Continuing process of combining equivalence classes

3 From Geometry to Topology

    Elastic deformations

    Intuitive idea of preservation of neighbourhoods

    Topological equivalence classes

    Derivation of 'topology'

    Close connection with study of continuity

4 Surfaces

    Surface of sphere

    "Properties of regions, paths and curves on a sphere"

    Similar considerations for torus and n-fold torus

    Separation of surface by curves

    Genus as a topological property

    Closed and open surfaces

    Two-sided and one-sided surfaces

    Special surfaces: Moebius band and Klein bottle

    Intuitive idea of orientability

    Important properties remain under one-one bicontinuous transformations

5 Connectivity

    Further topological properties of surfaces

    Connected and disconnected surfaces

    Connectivity

    Contraction of simple closed curves to a point

    Homotopy classes

    Relation between homotopy classes and connectivity

    Cuts reducing surfaces to a disc

    Rank of open and closed surfaces

    Rank of connectivity

6 Euler Characteristic

    Maps

    "Interrelation between vertices, arcs and regions"

    Euler characteristic as a topological property

    Relation with genus

    Flow on a surface

    "Singular points: sinks, sources, vortices, etc."

    Index of a singular point

    Singular points and Euler characteristic

7 Networks

    Netowrks

    Odd and even vertices

    Planar and non-planar networks

    Paths through networks

    Connected and disconnected networks

    Trees and co-trees

    Specifying a network: cutsets and tiesets

    Traversing a network

    The Koenigsberg Bridge problem and extensions

8 The Colouring of Maps

    Colouring maps

    Chromatic number

    Regular maps

    Six colour theorem

    General relation to Euler characteristic

    Five colour theorem for maps on a sphere

9 The Jordan Curve Theorem

    Separating properties of simple closed curves

    Difficulty of general proof

    Definition of inside and outside

    Polygonal paths in a plane

    Proof of Jordan curve theorem for polygonal paths

10 Fixed Point Theorems

    Rotating a disc: fixed point at centre

    Contrast with annulus

    Continuous transformation of disc to itself

    Fixed point principle

    Simple one-dimensional case

    Proof based on labelling line segments

    Two-dimensional case with triangles

    Three-dimensional case with tetrahedra

11 Plane Diagrams

    Definition of manifold

    Constructions of manifolds from rectangle

    "Plane diagram represenations of sphere, torus, Moebius band, etc. "

    The real projective plane

    Euler characteristic from plane diagrams

    Seven colour theorem on a torus

    Symbolic representation of surfaces

    Indication of open and closed surfaces

    Orientability

12 The Standard Model

    Removal of disc from a sphere

    Addition of handles

    Standard model of two-sided surfaces

    Addition of cross-caps

    General standard model

    Rank

    Relation to Euler characteristic

    Decomposition of surfaces

    "General classification as open or closed, two-sided or one-sided"

    Homeomorphic classes

13 Continuity

    Preservation of neighbourhood

    Distrance

    Continuous an discontinuous curves

    Formal definition of distance

    Triangle in-equality

    Distance in n-dimensional Euclidean space

    Formal definition of neighbourhood

    e-d definition of continuity at a point

    Definition of continuous transformation

14 The Language of Sets

    Sets and subsets defined

    Set equality

    Null set

    Power set

    Union and Intersection

    Complement

    Laws of set theory

    Venn diagrams

    Index sets

    Infinite

    Intervals

    Cartesian product

    n-dimensional Euclidean space

15 Functions

    Definition of function

    Domain and codomain

    Image and image set

    "Injection, bijection, surjection"

    Examples of functions as transformations

    Complex functions

    Inversion

    Point at infinity

    Bilinear functions

    Inverse functions

    Identity function

    "Open, closed, and half-open subsets of R "

    Tearing by discontinuous functions

16 Metric Spaces

    Distance in Rn

    Definition of metric

    Neighbourhoods

    Continuity in terms of neighbourhoods

    Complete system of neighbourhoods

    Requirement for proof of non-continuity

    Functional relationships between d and e

    Limitations of metric

17 Topological Spaces

    Concept of open set

    Definition of a topology on a set

    Topological space

    Examples of topological spaces

    Open and closed sets

    Redefining neighbourhood

    Metrizable topological spaces

    Closure

    "Interior, exterior, boundary"

    Continuity in terms of open sets

    Homeomorphic topological spaces

    Connected and disconnected spaces

    Covering

    Compactness

    Completeness: not a topological property

    Completeness of the real numbers

    "Topology, the starting point of real analysis"

  Historical Note

  Exercises and Problems

  Bibliography

  Index

Product Details

ISBN:
9780486419619
Author:
Flegg, H. Graham
Author:
Flegg, Graham
Author:
Flegg, H. Graham
Author:
Mathematics
Author:
H
Publisher:
Dover Publications
Location:
Mineola, NY
Subject:
Geometry - General
Subject:
Geometry
Subject:
Topology
Subject:
Topology - General
Subject:
General Mathematics
Subject:
MATHEMATICS / Topology
Copyright:
Edition Number:
1st Dover ed.
Edition Description:
Trade Paper
Series:
Dover Books on Mathematics
Series Volume:
no. WQ-95-74.
Publication Date:
20010931
Binding:
TRADE PAPER
Language:
English
Illustrations:
Yes
Pages:
208
Dimensions:
8.5 x 5.38 in 0.5 lb

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Related Subjects

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Science and Mathematics » Mathematics » Advanced
Science and Mathematics » Mathematics » General
Science and Mathematics » Mathematics » Geometry » Geometry and Trigonometry
Science and Mathematics » Mathematics » Topology
Young Adult » General

From Geometry to Topology New Trade Paper
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Product details 208 pages Dover Publications - English 9780486419619 Reviews:
"Synopsis" by ,
Introductory text for first-year math students uses intuitive approach, bridges the gap from familiar concepts of geometry to topology. Exercises and Problems. Includes 101 black-and-white illustrations. 1974 edition.
"Synopsis" by ,
This excellent introduction to topology eases first-year math students and general readers into the subject by surveying its concepts in a descriptive and intuitive way, attempting to build a bridge from the familiar concepts of geometry to the formalized study of topology. The first three chapters focus on congruence classes defined by transformations in real Euclidean space. As the number of permitted transformations increases, these classes become larger, and their common topological properties become intuitively clear. Chapters 4–12 give a largely intuitive presentation of selected topics. In the remaining five chapters, the author moves to a more conventional presentation of continuity, sets, functions, metric spaces, and topological spaces. Exercises and Problems. 101 black-and-white illustrations. 1974 edition.
"Synopsis" by ,
This introduction to topology eases readers into the subject by building a bridge from the familiar concepts of geometry to the formalized study of topology. Focuses on congruence classes defined by transformations in real Euclidean space, continuity, sets, functions, metric spaces, and topological spaces, and more. Exercises and Problems. Includes 101 black-and-white illustrations. 1974 edition.
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