Text: Elementary Topology

Dennis Roseman, Prentice-Hall, (1999)



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Who is the book for?

The purpose of this text is to provide an one semester introduction to topology at an advanced undergraduate, or beginning graduate level.

For some students this will be a first and last exposure to topology. If so, the student will leave with some interesting concrete examples, some interesting questions involving the plane and three dimensional space, and the impression that mathematical investigation of these matters is an interesting and useful endeavor.

For others, who will continue to learn about topology, our introduction will provide preparation for such basic texts as Munkres (Topology, a First Course), Willard (General Topology), or Sieradski (Introduction to Topology and Homotopy). For these students we provide a useful introduction that gives a background of good examples, yet minimizes duplication of future material.

For both groups of students we provide ample opportunity and guidance for building reasoning skills and writing proofs.

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How is this different from other books ?

The most common introduction to topology is through the study of general topology, either beginning with an axiomatic point of view, or perhaps with metric spaces. In any case, the approach is to present a slimmed version of general topology. The impression many students get from such a course is that topology is an abstract study of sets and subsets, many of which are not subsets of Euclidian space.

Our approach uses the setting of Euclidian space, mostly the line, the plane and 3-dimensional space. This provides an introduction, less abstract, than either of these others. This allows us to more quickly get to some interesting topics that one would not otherwise have time to consider. We focus questions such as: what are the different subsets of the plane (or space) and what should ``different'' mean?

Our text is ``example driven'' by considerations of subsets of n-space. As much as possible we attempt demonstrate the need for a tool before presenting it.

An additional benefit from this approach is that the study of Euclidian space interfaces with other mathematical areas familiar to the student. At various times we use geometry, analytic geometry, linear algebra, calculus, and complex numbers, to describe subsets and continuous functions. These connections show that topology is not isolated from things they already know about and also gives the students some familiar tools so they are not completely ``starting from scratch''.

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Features

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Table Of Contents

Note: the first 11 Chapters form the core material. Sections marked with asterisk are optional topics from the point of view of general topology.

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