110.413 INTRODUCTION TO TOPOLOGY 18 May 2005

Final Examination

by J. Michael Boardman, Department of Mathematics

Three hours. Open book, no notes. 100 points, 10 per question (plus an extra-credit challenge question). Partial credit may be available, but only if you show your working. Begin each question 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. The reals ℝ and ℝ^{2} are understood to have the standard topology. The integers are denoted by ℤ and the positive integers by ℤ_{+}.

1. Consider the collection of open balls B(x, 1/2^{n}) in ℝ^{2}, where x ∈ ℝ^{2} has rational coordinates and n is a positive integer. Prove that this collection is a basis of the standard topology on ℝ^{2}. (A complete formal proof is required here.)

2. Determine whether each of the following subsets of ℝ^{2} is open, closed, both, or neither. Give reasons. (Sketches may help.)

A = {(x, y) ∈ ℝ^{2} : x > 0 and y ≠ 0}; B = {(x, y) ∈ ℝ^{2} : x^{2} + y^{2} ≥ 2}; C = {(x, y) ∈ ℝ^{2} : x ∈ ℤ}; D = {(x, y) ∈ ℝ^{2} : x + y ∈ ℚ} (ℚ denotes the rationals); E = {(x, y) ∈ ℝ^{2} : x + y ≤ 0}.

3. Find which of the spaces in Question 2 are connected and which are pathconnected. Give reasons. (Again, sketches may help.)

4. Let p: X → Y be a surjective closed continuous map such that p^{−1}(y) is compact for each y ∈ Y. If X is Hausdorff, show that Y is Hausdorff. (Recall that the map p is closed if p(C) is closed in Y whenever C is closed in X.)

5. Is each of the following sets finite, countably infinite, or uncountable? Give reasons. If finite, find an upper bound for its cardinality.

F, the set of all equivalence relations on the set ℤ; G, the set of all rationals of the form m/2^{n}, where m and n are positive integers; H, the set of all functions f: ℤ_{+} → ℤ_{+} that satisfy f(n) = n for all sufficiently

large n (how large depends on f); J, the set of all topologies on a finite set with n elements; K, the set of all (simple) order relations on the set ℤ.

OVER →

6. Which of the following spaces are simply connected? Give reasons.

L, the unit interval [0, 1] in ℝ;

M, the subspace of all x ∈ ℝ^{2} that satisfy ∥x∥ ≥ 2;

N, the subspace of all (x, y) ∈ ℝ^{2} that satisfy y ≠ 0.

7. Let X and Y be metric spaces.

(a) Find a metric on X × Y that induces the product topology on X × Y, and prove that it does.

(b) Does your metric make the projection map π_{1}: X × Y → X uniformly continuouos? Justify your answer.

8. Show that a connected normal space that contains more than one point is uncountable. [Hint: The book’s definition (not universally accepted) of normal includes the condition that each singleton {x} is closed.]

9. Let B be a basis of the topological space X. Show that X is compact if and only if each covering of X by members of B has a finite subcovering. (Again, a real formal proof is required.)

10. Find the interior, boundary and closure of each of the following subsets: P = {x ∈ ℝ^{2} : 1 ≤ ∥x∥ < 2}, a subset of ℝ^{2}; Q, the set of all irrational points in the unit interval [0, 1], considered as a subset

of ℝ.

11. (extra-credit challenge question) Given (simply) ordered sets X and Y, find necessary and sufficient conditions for the dictionary order topology on X × Y to coincide with the product topology. (Assume that X and Y both contain at least three points.)

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