7.14. EXERCISES 175

(R,ρ) is not a complete metric space while (R,d) is. Thus, unlike compactness.Completeness is not a topological property. Hint: To show the lack of completenessof (R,ρ) , consider xn = n. Show it is a Cauchy sequence with respect to ρ .

17. A very useful idea in metric space is the following distance function. Let (X ,d) be ametric space and S⊆ X ,S ̸= /0. Then dist(x,S)≡ inf{d (x,y) : y ∈ S} . Show that thisalways satisfies

|dist(x,S)−dist(z,S)| ≤ d (x,z)

This is a really neat result.

18. If K is a compact subset of (X ,d) and y /∈ K, show that there always exists x ∈ Ksuch that d (x,y) = dist(y,K). Give an example in R to show that this is might not beso if K is not compact.

19. You know that if f : X → X for X a complete metric space, then if d ( f (x) , f (y))<rd (x,y) it follows that f has a unique fixed point theorem. Let f : R→ R be givenby

f (t) = t +(1+ et)−1

Show that | f (t)− f (s)|< |t− s| , but f has no fixed point.

20. If (X ,d) is a metric space, show that there is a bounded metric ρ such that the opensets for (X ,d) are the same as those for (X ,ρ).

21. Let (X ,d) be a metric space where d is a bounded metric. Let C denote the collectionof closed subsets of X . For A,B ∈ C , define

ρ (A,B)≡ inf{δ > 0 : Aδ ⊇ B and Bδ ⊇ A}

where for a set S,

Sδ ≡ {x : dist(x,S)≡ inf{d (x,s) : s ∈ S} ≤ δ} .

Show x→ dist(x,S) is continuous and that therefore, Sδ is a closed set containing S.Also show that ρ is a metric on C . This is called the Hausdorff metric.

22. ↑Suppose (X ,d) is a compact metric space. Show (C ,ρ) is a complete metric space.Hint: Show first that if Wn ↓W where Wn is closed, then ρ (Wn,W )→ 0. Now let{An} be a Cauchy sequence in C . Then if ε > 0 there exists N such that whenm,n≥ N, then ρ (An,Am)< ε. Therefore, for each n≥ N,

(An)ε⊇ ∪∞

k=nAk.

Let A≡ ∩∞n=1∪∞

k=nAk. By the first part, there exists N1 > N such that for n≥ N1,

ρ(∪∞

k=nAk,A)< ε, and (An)ε

⊇ ∪∞k=nAk.

Therefore, for such n, Aε ⊇Wn ⊇ An and (Wn)ε⊇ (An)ε

⊇ A because

(An)ε⊇ ∪∞

k=nAk ⊇ A.