176 CHAPTER 7. METRIC SPACES AND TOPOLOGICAL SPACES

23. ↑ Let X be a compact metric space. Show (C ,ρ) is compact. Hint: Let Dn be a 2−n

net for X . Let Kn denote finite unions of sets of the form B(p,2−n) where p ∈ Dn.Show Kn is a 2−(n−1) net for (C ,ρ) .