MAT 327 Homework assignment 3. Due Thursday, October 29, 2015 Each problem is worth 10 points Problem 1. Give an example of a metric space X and a closed and bounded subspace Y of X that is not compact. Problem 2. Let f : X Y be a map from a topological space X to a compact Hausdor space Y . Show that if the graph of f is closed in X Y , then f is continuous. Problem 3. Suppose that X and Y are locally compact, Hausdor topological spaces, f : X Y is one-to-one, onto, continuous and proper (preimages of compact sets are compact). Prove that f is a homeomorphism. Problem 4. Let X be a compact Hausdor space. Let K1 K2 ... be a sequence of closed connected subsets of X. Let K = Ki . Show that K is i=1 connected. Problem 5. Let (K, d) be a compact metric space, and f : K K be a map such that d(f (x), f (y)) < d(x, y) for all x = y. Denote K0 = K and dene recursively Ki+1 = f (Ki ). Prove that Ki is a one-point set. i=1 Problem 6. Let A be a subset of lR2 with the euclidean metric, satisfying the property that every continuous function f : A lR has a maximum in A. Prove that A is compact. MAT 327 Homework assignment 3. Due Thursday, October 29, 2015 Each problem is worth 10 points Problem 1. Give an example of a metric space X and a closed and bounded subspace Y of X that is not compact. Problem 2. Let f : X Y be a map from a topological space X to a compact Hausdor space Y . Show that if the graph of f is closed in X Y , then f is continuous. Problem 3. Suppose that X and Y are locally compact, Hausdor topological spaces, f : X Y is one-to-one, onto, continuous and proper (preimages of compact sets are compact). Prove that f is a homeomorphism. Problem 4. Let X be a compact Hausdor space. Let K1 K2 ... be a sequence of closed connected subsets of X. Let K = Ki . Show that K is i=1 connected. Problem 5. Let (K, d) be a compact metric space, and f : K K be a map such that d(f (x), f (y)) < d(x, y) for all x = y. Denote K0 = K and dene recursively Ki+1 = f (Ki ). Prove that Ki is a one-point set. i=1 Problem 6. Let A be a subset of lR2 with the euclidean metric, satisfying the property that every continuous function f : A lR has a maximum in A. Prove that A is compact