110.413 INTRODUCTION TO TOPOLOGY 2 March 2005

Mid-Term Examination

by J. Michael Boardman, Department of Mathematics

50 minutes. 40 points, 10 per question. THIS IS AN OPEN BOOK EXAMINATION. Partial credit may be available, but only if you show your working. Begin each of the four questions on a new page and number it clearly in the margin. Don’t forget to put your name on the cover of the book. Use only the officially provided blue books.

1 Decide whether each of the following sets is finite, countably infinite, or uncountable. Give reasons.

(a) The set of all circles in the plane ℝ^{2} that have integer radius;

(b) The set ℚ × ℤ;

(c) The set of all order-preserving bijections ℤ_{+} → ℤ_{+};

(d) The set of all order-preserving functions ℤ_{+} → ℤ_{+};

(e) The set of all order-preserving bijections ℤ → ℤ.

2 Define the relation ∼ on the plane ℝ^{2} = ℝ × ℝ by:

(x_{1}, y_{1}) ∼ (x_{2}, y_{2}) if and only if x_{1} + y_{1} = x_{2} + y_{2} + n for some integer n.

Prove that ∼ is an equivalence relation. Describe a typical equivalence class for this relation.

3 On the plane ℝ^{2}, define

d((x_{1}, y_{1}), (x_{2}, y_{2})) = max{|x_{2} − x_{1}|, |x_{2} + y_{2} − x_{1} − y_{1}|}

Prove that d is a metric on ℝ^{2}. Sketch the ball B((0, 0), 2).

4 Give the real line ℝ the standard ordering.

(a) Give an example of a non-empty subset of ℝ that is bounded above but has no maximum (or greatest) element.

(b) Give an example of a proper subset of ℝ that is not bounded above.

(c) Give an example of an ordered set X and a non-empty subset A that is bounded above but has no least upper bound.

(d) Give an example of a non-empty ordered set in which every element has an immediate predecessor and an immediate successor.

(e) Find three order relations on the set ℤ_{+} that have different order types.

YOU MAY RETAIN THIS QUESTION SHEET