Overview
Problem 1. Exercises 1.5.4, 1.5.5, 2.1.1, 2.1.4, 2.3.4, 2.4.6, 2.4.7 from the book.
Problem 2. Let (X; dX) and (Y; dY ) be two metric spaces. A function f : X ! Y is said to be Lipschitz
continuous with Lipschitz constant L > 0 if dY (f(x); f(x0)) LdX(x; x0) for all x; x0 2 X. Show that if f
is Lipschitz continuous, then f is uniformly continuous.
Problem 3. Show that the connected components of Q are the singletons. In other words, Q has no
nontrivial connected subsets. (Such a space is also called totally disconnected.)
Hint: Suppose E Q contains two di erent points x < y. Use the fact that there exists an irrational
number a such that x < a < y to show that E is disconnected.