110.413 INTRODUCTION TO TOPOLOGY 14 October 1998

Mid-Term Examination

50 minutes. Open book. 60 points, 20 per 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.

1. Given a metric space (X, d), define

d′(x, y) = \frac{d(x, y)}{1 + d(x, y)}

for all x, y ∈ X. Show that d′ is a metric on X that defines the same topology as d.

2. Give R^{2} the euclidean metric, and the resulting standard topology. Which of the following collections of subsets are bases of the standard topology? In each case, give reasons.

(a) All open disks B((x, y), r), where x, y, and r are required to be rational numbers whose denominators are powers of 2.

(b) All open squares (a, b) × (c, d), where a, b, c, and d are required to be integers.

(c) All open disks B((x, y), r), where we require r < 1/100.

(d) All open disks B((a/n, b/n), 1/n^{4}), where a, b and n are integers (n > 0).

3. Let A and B be any subsets of a topological space X.

(a) Is \overline{A ∩ B} = \overline{A} ∩ \overline{B}?

(b) Is \overline{A − B} = \overline{A} − \overline{B}? In each case, give either a proof or a counterexample.

YOU MAY RETAIN THIS QUESTION SHEET JMB