ASSIGNMENT 6
MAT 472 ·FALL 2005
Problem 1. Let (E, dE) and (F , dF) be separable metric spaces. Prove that (E×F, d) is
separable, where dis the metric on E×Fdefined by
d((e, f ),(e0, f0)) = max{dE(e, e0), dF(f , f0)}.
A metric space (E, d) is totally bounded if for each > 0 there exists a finite set F⊆E
such that
E=[
p∈F
B(p).
Problem 2. Prove that every totally bounded metric space is separable.
Problem 3 (See Problem III.36).Suppose (E, d) is a metric space which is not totally
bounded. Prove that there exists a sequence in Ewhich has no Cauchy subsequence.
Date: September 29, 2005 / Due Date : Tuesday, October 4, 2005.
S. Kaliszewski, Department of Mathematics and Statistics, Arizona State University.
1