MATH 214 Final, Fall 1996

op bats Math 214 | Ke Final Exam February 1, 1997 1. Let D be a dense subset of a space X and U an open subset of X. Prove that D(\u =u. 2. Let X and Y be two topological spaces, A a subset of X and B a subset of Y. Show that Ax B=AxB. 3. Let f:X — Y be a map from a space X to a space Y and G, = {(x, Ax)) e X x Y | x eX} the graph of f Prove that the mapping g : X >G, given by g(x) = (x, &{(x)) is a homeomorphism from the space X to the subspace G; of X x Y if and only if the map fis continuous. 4. Let (Aj);. ; be a family of connected subsets of a space X such that A; QA; # @ for any i, je |. Show that UA is a connected subset of the space X. i 5. Prove that a space X’is regular if and only if the following condition holds : For every closed subset A of X and every point x ¢ A, there exists a nbhd. U of x such that UNA=@. Formulate an analogous statement that is equivalent to normality.