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TopologyMathematics

For a onesemester, advanced undergraduate level course
in Introduction to Topology. Designed both for students who will
take only one course in topology as well as for those who are preparing
for more advanced work, this text offers a thorough introduction to
the important topics of topology, a variety of interesting, concrete
examples, and ample opportunity and guidance for building reasoning
skills and writing proofs. It integrates students' background in calculus,
analytic geometry and linear algebra throughout the presentation.
Presents a selection of interesting advanced topics
— e.g., manifolds, complexes, knots— in a manner undergraduates can handle.
 Some basic mathematical results derived from
the basic core material are included.
Provides an extra, “guided index” — an
extensive list of topics collected as Examples, Propositions and Remarks
— designed to help students solve problems and investigate examples.
Features a lessabstract, more exampledriven
approach that uses the setting of Euclidean space — mostly
the line, the plane and 3dimensional space.
 Focuses on questions, e.g.: What are the different subsets
of the plane (or space) and what should “different” mean?
 Considers subsets of Rn. Demonstrates the
need for a tool before presenting it.
 Uses geometry, analytic geometry, linear algebra, calculus,
and complex numbers to describe subsets and continuous functions —
and to show that topology is not isolated from things about which students already
know.
Focuses on Rn — to get more quickly and more
indepth into more interesting problems (rather than getting into
long discussions of separation properties, bases and subbases, countability).
 Presents topics of general topology — not relevant
to Rn — in an appendix.
Emphasizes examples.
 Develops concepts to answer questions about some subsets
of Euclidean space.
 Several proofs are given to verify a single relationship.
Explores basic mathematical questions — e.g.,
What are the subsets of R2? What are the possibilities
of embedding one object into another?
Offers some novel alternate approaches to standard problems,
e.g.:
 The first proof offered that R1 is not homeomorphic
to R2 is that there are nonequivalent embeddings of a
three point set into R1, but all embeddings of a three
point set into R2 are equivalent.
 The first proof that the interval, I, is not
homeomorphic to the circle, S1, is that I has
the fixed point property, S1 does not.
Preface
I. BASIC TOPICS.
1. Open and Closed Subsets.
2. Building Open and Closed Subsets.
3. Continuity.
4. Homeomorphism.
5. Cantor Sets and Allied Topics.
6. Embeddings.
7. Connectivity .
8. Path Connectedness.
9. Closure and Limit Points.
10. Compactness.
11. Local Connectivity.
II. ADVANCED TOPICS
12. Space Filling Curves.
13. Manifolds.
14. Knots and Kottings.
15. Simple Connectivity.
16. Deformation Type.
17. Complexes.
18. Higher Dimensions.
19. The Poincaré Conjecture.
III. APPENDICES.
Appendix A. Sets and Logic.
Appendix B. Numbers.
Appendix C. Cardinality of Sets.
Appendix D. Summary from Calculus.
Appendix E. Strategy in Proof.
Bibliography.
Index of Examples, Remarks, and Propositions.
Subject Index.
