12 CONTENTS a metric on the set X that induces the topology of X Uses of the book We taught this manuscript in the first year PhD core sequence at the Uni- versity of Pittsburgh and as a Phd class at the University of Texas. The program at University of Pittsburgh begins with an intensive, one month re- medial summer math class that focuses on calculus and linear algebra. Our manuscript was used in the Fall semester class. Since we were able to quickly explain theorems using sketches of proofs, it was possible to teach the entire book in one semester. If the book was used for upper level undergradu ates, we would suggest simply to teach Chapters 1 to 4. While we used the manuscript in a classroom, we expect it will be beneficial to researchers; for instance, anyone who reads a book like Stokey and Lucas' Recursive Meth- ods must understand the background concepts in our manuscript. In fact, it was because one of the authors found that his students were ill prepared to understand Stokey and Lucas in his upper level macroeconomics class, that this project began12 CONTENTS a metric on the set X that induces the topology of X). Uses of the book We taught this manuscript in the first year PhD core sequence at the Uni￾versity of Pittsburgh and as a PhD class at the University of Texas. The program at University of Pittsburgh begins with an intensive, one month re￾medial summer math class that focuses on calculus and linear algebra. Our manuscript was used in the Fall semester class. Since we were able to quickly explain theorems using sketches of proofs, it was possible to teach the entire book in one semester. If the book was used for upper level undergradu￾ates, we would suggest simply to teach Chapters 1 to 4. While we used the manuscript in a classroom, we expect it will be beneficial to researchers; for instance, anyone who reads a book like Stokey and Lucasí Recursive Meth￾ods must understand the background concepts in our manuscript. In fact, it was because one of the authors found that his students were ill prepared to understand Stokey and Lucas in his upper level macroeconomics class, that this project began