Eksamen i Matematik 3GT Januar 2003

Matematik 3GT

Books, notes, calculators, and computers are allowed at this 3 hour written exam. You may write your answers in pencil.

Problem 1 (10 points)

Let X be a strictly simply ordered set. Assume that X is compact in the order topology. Show that X has a smallest and a largest element.

Problem 2 (40 points)

Let X be a compact metrizable space.

(1) Show that X contains a dense countable subset A. (Hint: Cover X by finitely many 1/n-balls.)

(2) Show that the topology on X is induced by a metric d: X × X → [0, 1] with values in the interval [0, 1].

(3) Let A ⊂ X be as in (1) and the metric d as in (2). Show that the map

f: X → [0, 1]^{A} given by f(x) = (d(x, a))_{a∈A}

is an imbedding and that the image f(X) ⊂ [0, 1]^{A} is closed.

(4) Show that any closed subspace of [0, 1]^{ω} is compact and metrizable.

Problem 3 (20 points)

For a locally compact Hausdorff space X, let X^{•} = X ∪ {∞} denote its one-point-compactification. For a continuous function f: X → Y between two such spaces, let f^{•}: X^{•} → Y^{•} be the extension of f given by f(∞) = ∞.

(1) Show that f^{•}: X^{•} → Y^{•} is continuous if and only if f^{−1}(K) is compact for any compact subspace K ⊂ Y.

(2) Is g^{•}: R^{•} → (R^{2})^{•} continuous when g: R → R^{2} is the continuous map g(t) = (cos t, sin t)?

Problem 4 (50 points)

Let P^{2} = S^{2}/ ∼ be the quotient space of the unit 2-sphere

S^{2} = {(x_{1}, x_{2}, x_{3}) ∈ R^{3} | x_{1}^{2} + x_{2}^{2} + x_{3}^{2} = 1} ⊂ R^{3}

by the equivalence relation x ∼ y ⇔ x = ±y. The quotient map, sending a point to its equivalence class, is denoted p: S^{2} → P^{2}.

(1) Show that the map p is open and closed.

(2) Show that P^{2} has a countable basis.

(3) Show that P^{2} is Hausdorff.

(4) Let U = {(x_{1}, x_{2}, x_{3}) ∈ S^{2} | x_{3} > 0} ⊂ S^{2} be the upper open hemisphere. Show that p|U: U → P^{2} is an imbedding.

(5) The complement P^{2} − p(U) is a familiar topological space. Which one? (Just give the answer, no justification is necessary.)

(THE END)