Homework 5
Math 447: Real Variables
Exercise 1 Prove that any inite intersection of open sets is open: if (X; d) is any metric space and 
are open sets in X then 
is open in X .
Exercise 2 Let (X; d) be a metric space and (xn) a sequence in (X; d). Prove that Lim(xn)c is an open set in X .
Exercise 3 Let p ∈ N be any prime number. For each nonzero rational number r there is a unique k such that 
such that a, b ∈ Z and p divides neither a nor b.
For example, if p = 3 we may write 12 = 31 · 4, 20 = 30 · 20, and 7/18 = 3-2 · 7/2.
Deine
In our example, |12|3 = 3- 1 |20|3 = 30 = 1 and |7/18|3 = 32. Similarly, |12|5 = 1 = |7/18|5 and |20|5 = 5-1. Prove that the function dp (m, n) = |m — n|p makes (Q, dp) into a metric space.
Exercise 4 For the metric dp deined in Exercise 3, describe each of the sets below using set notation and list three elements of each set. Your inal description should not use the metric directly (so answers similar to {x : |x|p ≤ 1} are not valid).
(a) The open ball B1 (0).
(b) The closed ball B1 (0).
(c) The open ball Bδ (0) with δ = p-k.