advertisement

Math 434/Math 542 Test 1 Carter Fall 2012 One heaping (1 ounce) scoop of coffee for each 8-10 ounce cup, plus 1/2 for the pot. Freshly ground is better. I prefer a dark roast. 1. According to Wikipedia the topologies on a 3 element set are (up to homeomorphism — which are induced by permutations) the following: (a) {∅, {a, b, c}} (b) {∅, {c}, {a, b, c}} (c) {∅, {a, b}, {a, b, c}} (d) {∅, {c}, {a, b}, {a, b, c}} (e) {∅, {c}, {b, c}, {a, b, c}} (f) {∅, {c}, {a, c}, {b, c}, {a, b, c}} (g) {∅, {a}, {b}, {a, b}, {a, b, c}} (h) {∅, {b}, {c}, {a, b}, {b, c}, {a, b, c}} (i) {∅, {a}, {b}, {c}, {a, b}, {a, c}, {b, c}{a, b, c}} Determine when the identity map is continuous. 2. Let X denote a set. Let A 7→ A denote a function P(X) → P(X) defined on the power set of X that satisfies the following rules: (a) A ⊂ A for any A ∈ P(X) (b) A = A (c) A ∪ B = A ∪ B (d) ∅ = ∅ Define a set to be closed if A = A. Show that this defines a collection of subsets that is closed under arbitrary intersection and finite union. 3. Prove that any continuous function f : [0, 1] → [0, 1] has a fixed point (i.e. there is a point x ∈ [0, 1] such that f (x) = x). Hint: You may assume that 0 < f (0). 4. Show that the set of all functions {f : Z+ → {0, 1}} cannot be put into one-to-on correspondence with the positive integers: Z+ = {1, 2, 3, . . .}. Such a function is an infinite sequence of 0s and 1s. 5. Give an inductive proof that the number of functions from an n element set ({1, 2, . . . , n}) to an m element set {1, 2, . . . , m} is mn .