110.413 INTRODUCTION TO TOPOLOGY 12 May 2004

Final Examination

by J. Michael Boardman, Department of Mathematics

Three hours. Open book. No notes. 100 points, 10 per question. Partial credit is available, but answers unsupported by good reasons will be worth

little.

The spaces ℝ, ℝ^{2} and ℚ are always given the standard topology. Begin each of the ten questions on a new page and number it clearly in the margin. If you use two books, label them “Book 1 of 2” and “Book 2 of 2”. (If you use

three books, …)

Use only the officially provided blue books.

1. Let A ⊂ X and B ⊂ Y be subspaces. In the space X × Y, prove that \overline{A × B} = \overline{A} × \overline{B}.

2. Decide whether each of the following sets is finite, countably infinite, or uncountable:

The set A of all square-free positive integers. (An integer is square-free if it is not divisible by n^{2} for any integer n > 1.)

The set B of all finite subsets of the real line ℝ. The set C of all lines in the plane ℝ^{2} that have rational slope. The set D of all subsets of the set of all subsets of the set {1, 2, 3, 4, 5, 6, 7, 8, 9}.

3. Let A be a connected subspace of a space X.

(a) Is the interior Int A of A necessarily connected?

(b) Is the closure \overline{A} of A necessarily connected?

(c) Is the boundary Bd A of A necessarily connected?

4. Let (X, d) be a metric space.

(a) If x_{0} ∈ X, show that the function f: X → ℝ defined by f(x) = d(x_{0}, x) is uniformly continuous.

(b) Given a nonempty subset A of X, show that the function g: X → ℝ defined by g(x) = d(A, x) is uniformly continuous, where d(A, x) = inf_{y∈A} d(y, x).

5. Which of the following spaces are simply connected?

(a) The subspace ℝ^{2} − {(0, 0)} of the plane ℝ^{2};

(b) The real line ℝ;

(c) The ball B(0, r) radius r in the plane ℝ^{2}.

OVER →

6. Let (X, d) and (Y, d) be complete metric spaces.

(a) Give a metric for the space X × Y, and prove that it induces the product topology.

(b) Is your metric on X × Y complete?

7. Let A and B be compact subspaces of the Hausdorff space X. Are the spaces A ∩ B and A ∪ B necessarily compact?

8. Decide whether the following subsets of the plane ℝ^{2} are open, closed, both, or neither. [Sketches may help.]

(a) The set of all (x, y) that satisfy xy = ±1;

(b) The set of all (x, y) for which x is rational and |y| ≤ 1;

(c) The set of all (x, y) that satisfy x^{2} + y^{2} = −2;

(d) The set of all (x, y) in which x is an integer;

(e) The set of all (x, y) with y^{2} > 2.

9. Let p: X → Y be a closed surjective continuous map, where X is normal. (p is a closed map if p(C) is closed in Y whenever C is closed in X.) Prove that Y is normal.

10. Which of the following spaces are connected? Give a separation when there is one. [Sketches may help.]

(a) The set E of all points (x, y) in the plane ℝ^{2} that satisfy xy = 4;

(b) The set F of all points (x, y) in the plane ℝ^{2} that satisfy 4x^{2} + y^{2} = 1;

(c) The set ℚ of all rational numbers;

(d) The set G of all points (x, y) in the plane ℝ^{2} that satisfy y = ±x.

YOU MAY RETAIN THIS QUESTION SHEET