MAT 530 (Fall 1995) Topology/Geometry I

Text and References

John G. Hocking and Gail S. Young, Topology, Dover Publications, Inc., New York 1988. (This work was first published, by Addison-Wesley, in 1961)
This is an excellent book that gives the motivation for topological concepts along with rigorous definitions, and does a good job of communicating why topologists love topology. Some of the terminology is somewhat archaic. The concept of category appears only implicitly in the first section. The authors use transformation where today one more commonly hears map or mapping, and separated for our disconnected (beware a possible confusion with the French usage of séparé to mean Hausdorff !).


Nicolas Bourbaki, General Topology

John Kelly, General Topology, Van Nostrand, Princeton NJ 1955.
Includes all you ever wanted to know about the Axiom of Choice and more. This book is a model of elegant exposition; you know you're in the hands of a master.

James Munkres, Topology, a First Course, Prentice Hall, Englewood Cliffs NJ 1975

Leopoldo Nachbin, The Haar Integral, Van Nostrand, Princeton NJ 1965.
Useful to us for the transparent proof of the Tychonoff Theorem.

André Weil, Sur les théorèmes de de Rham, Commun. Math. Helv. (1952) 119-145
This paper, which gives proofs of de Rham's theorems relating the de Rham, singular and Cech homology and cohomology for smooth manifolds, is constructed like a beautiful watch. There is not a superfluous comma, and never an inelegant turn of phrase. A low-key conversation with a mathematical giant. (The proofs, organized by double complexes, are the modern sheaf-theoretic proofs avant la lettre. Everything is there but the terminology, and everything is completely explicit.)

Hassler Whitney, Annals of Math. 37 (1936) 668-672
Thanks to Larry Cruvant for this reference.

Hassler Whitney, Geometric Integration Theory, Princeton Univ. Press, Pronceton New Jersey 1957.