MATH 214 Final, Fall 2000

January 30, 2001 Mathematics 214, First Semester 2000-2001 Final Examination. Problem 1. & pts. Whal do von know about topological iivariants? Write the definition aud Jist as many topological iivariants as yor know. Then pive two different examples of how Copological iivariants cag be used. Problem 2. & pls. Let XY bea set and fyg: XN o> XN he two fianetious. Suppose that fog = idy, that is, that f(g(2)) = 2 for cach 2 EN. Prove that fois oulo aud that g is onc-bo-one, Problem 3. 8 pts, Let XN be a anetrizable space. Let A CX be a non-einpty set and cousider A with its subspace topology inherited from X. Prove that fo N is separable, then v4 is separable, too. Problem J. & pts. Let XY and Yo be two topological spaces Prove that the product space Xx Y is separable if and only if both XN and Y are separable. Problem 5, & pts Show (hat ne two of the following tree spaces are homeomorphic: R, Re and the unit circle S = {Cr,y) € Resa +4 = 1} (each space is with ils usnal topology) Bonus, (pis. Let XY be a topologies) space. Consider the prodnet space Nx X and its subsct A= {(r.2): 2 € X} Qnost offen, Ais called the diagonal). Show that if Ais an opeu set in Vo AX, then X is a discrete space, Good Luck! Time: 120 minutes.