MATH 214 Final, Fall 2001

ERICAN CNIVERSITS® LIBRARY OF eines February 2, 2002 Mathematics 214, First Semester 2001-2002 Final Examination. Problem 1. 10 pts. What do you know about metrizable spaces? Problem 2. 6 pts. Prove that each compact metrizable space is separable. Problem 3. 6 pts. Prove that if X and Y are regular spaces, then X x Y is a regular space, too. Problem 4. 3+ 6 pts. Consider subspaces of R? represented by geometric shapes of the capital letters A, E, P and H. (a) Are any two of them homeomorphic? If “yes”, which ones? (b) Choose 3 non-homeomorphic pairs among them and prove that each of your choices is correct. Problem 5. 6+ 3 pts. Let X be a Hausdorff space. Suppose that f : X — X is a continuous function such that fo f = f. Let A= f(X). (a) Prove that A= {x € X: f(z) =z}. (b) Show that A is a closed subset, of X. AMERICAN UNTeERe re LIBRARY OF BerRey Bonus. 5 pts. Let X be a countable metrizable space. Prove that X is totally disconnected. Does it have to be zero- dimensional? Good Luck! Time: 120 minutes. ERICAN UNIVERSIT®) LIBRARY OF keimEY