This book presents an introduction to topology that begins with the axiomatic definition of a topology on a set, rather than starting with metric spaces or the topology of subsets of Rn. This approach includes many more examples, allowing students to develop more sophisticated intuition and enabling them to learn how to write precise proofs in a brand-new context, which is an invaluable experience for math majors. Along with the standard point-set topology topics Topology covers the construction of spaces from other spaces, including products and quotient spaces. This innovative text culminates with topics from geometric and algebraic topology (the Classification Theorem for Surfaces and the fundamental group), which provide instructors with the opportunity to choose which "capstone" best suits his or her students.
· Introduction: Intuitive Topology.
· Background on Sets and Functions.
· Topological Spaces.
· More on Open and Closed Sets and Continuous Functions.
· New Spaces from Old.
· Connected Spaces.
· Compact Spaces.
· Separation Axioms.
· Metric Spaces.
· The Classification of Surfaces.
· Fundamental Groups and Covering Spaces.
As a textbook for a junior/senior level topology course, intended to be covered in one semester by typical math majors (as opposed to some "one-semester" texts that are so dense that typical classes can cover only a small portion in one term); as a reference guide for graduate students; university libraries.
Paul L. Shick, PhD, is a Professor in the Department of Mathematics and Computer Science at John Carroll University in Cleveland, OH. He has authored numerous journal articles and is a member of the American Mathematical Society, the Mathematical Association of America, and the American Association of University Professors. Dr. Shick received his PhD in 1984 from Northwestern University.