MATH 214 Final, Fall 2004

January 19, 2005 Mathematics 214, First Semester 2004-2005 Final Examination. Problem 1. 12 pts. What do you know about compact spaces? Remind the definition. Give examples of compact and non-compact metrizable spaces. Then give examples of compact and non-compact spaces that are Hausdorff but not metriz- able. Finally, quote at least 6 theorems about compact spaces. Problem 2. 6 pts. (This is a “set theory question”.) Let X and Yj, Yo,...,¥, be sets and let f : X — ¥, x... x ¥, be a function such that f(t) = (filz), fo(x),...,fn(v)) for each x € X. Suppose that A; CY; fori=1,2...,n. Prove that f~!(A1 x Ap x... x An) = fp (Ar) fy Ao) 9... f7t(An) - Problem 3. 6 pts. Let X be a topological space, U’ be an open subset of X, and D be a dense subset of X. Prove that UA D=U . Problem 4. 6 pts. Let X be a topological space and AC X. Let {0,1} be a two-point discrete space. Let f : X — {0,1} be given by the rules that f(z) =1 when x € A, and f(x) = 0 when « ¢ A. Prove that f is continuous if and only if A is a closed-open set in X. Problem 5. 6 pts. Give a detailed outline of a proof that each subspace of a compact metrizable space X must be separable. Do not provide technical details. Problem 6. 6 pts. Prove that if X x Y is a normal space, then X is a normal space. Bonus. 4 pts. Give an example of a continuous function f : R* — R such that F(R) = (0,1). Good Luck! Time: 150 minutes. Taste has fo Se P LER RARY | oe URE nent camara aosarwntn