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 .