Math 335 -- Topology I 

Topology I is the first semester of topology, taken by mathematics majors to complete their education in pure mathematics (the major branches of which are real and complex analysis, abstract algebra, and geometry and topology), and by mathematically adventurous students in other disciplines who are interested in its applications to fixed point theorems, for example, in economics; its applications to cosmology, general relativity and string theory in physics; or knotted structures like DNA or biological clocks.

In addition to a thorough coverage of point set topology, to which students are introduced in Math 317, it provides an introduction to algebraic topology. The fundamental group and covering spaces are discussed in depth to prepare students for either Topology II at Haverford or a graduate course in Topology at Bryn Mawr. In recent years, Topology II at Haverford has covered simplicial homology theory (since graduate programs teach the less concrete singular homology theory), but if students are interested it could cover differential geometry instead. Along with Real and Complex Analysis and two semesters of Abstract Algebra, a one semester course in Topology is ideal for students applying to graduate programs in mathematics.

Prerequisites: Math 317 and Math 333. With permission, either may be taken as a corequisite.

Who should take the course:
Students majoring or minoring in mathematics who want to see the role of analysis and algebra in the third branch of pure mathematics.
Students interested in applications of topology to equilibrium theory in economics, astronomy and general relativity, or biology.

Topics covered:
Set theory and logic, including the axiom of choice, the well-ordering theorem and transfinite induction
Topological spaces and continuous functions, including the pasting lemma and the quotient topology
Connectedness and compactness, including the tube lemma and the Lebesque number lemma
The fundamental group and covering spaces, including the Brouwer fixed point theorem and the Jordan curve theorem


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For detailed information about Math 335 this year, please consult the list of Fall Courses and Spring Courses linked to the Mathematics and Statistics Home Page.