Math 446 Homework 6, due Monday, October 30, 2017

(1) Let (X, d) be a metric space. Then, as we saw on homework 3,the function ρ : X ×X → R, defined by

ρ(x, y) :=d(x, y)

1 + d(x, y),

is also a metric on X.(a) Prove that d and ρ are uniformly equivalent metrics on X.(b) Prove that (X, d) is complete if and only if (X, ρ) is com-

plete.(c) Prove that every metric space (X, d) is homeomorphic to

a metric space (Y, dY ) of diameter at most 1.

Products of two metric spaces: The product of two metric spaces(Y, dY ) and (Z, dZ) is the metric space (Y × Z, dY×Z), where dY×Z isdefined by dY×Z((y, z), (y′, z′)) = dY (y, y′) + dZ(z, z′).

(2) (a) Let ((yn, zn))n∈N be a sequence in Y × Z and let (y, z) ∈Y ×Z. Prove that ((yn, zn))n∈N converges to (y, z) in (Y ×Z, dY×Z) if and only if (yn)n∈N converges to y in (Y, dY )and (zn)n∈N converges to z in (Z, dZ).

(b) Assume that (Y, dY ) and (Z, dZ) are compact. Prove that(Y × Z, dY × dZ) are compact.

(c) Let (X, dX) be a metric space. Prove that the function dX :X×X → R is continuous as a function from (X×X, dX×X)to (R, d|·|).

Products of sequences of metric spaces: Let (X1, d1), (X2, d2), . . .be a sequences of metric spaces. Then their product is the metric spae(X, ρ), defined as follows: First, we replace each of the metrics dn byan equivalent metric ρn of diameter at most 1 (problem (1) shows howthis can be done). We let X =

∏∞k=1Xk denote the Cartesian product

of the sets X1, X2, . . . , i.e., X is the set of all functions f , with domainN, and with f(k) ∈ Xk for all k ∈ N. Then, the metric ρ on X isdefined by

ρ(f, g) =∞∑k=1

2−kρk(f(k), g(k)).

We saw in homework 3 that ρ is indeed a metric on X.

(3) (a) Let (fn)n∈N be a sequence in X and let f ∈ X. Provethat (fn)n∈N converges to f in (X, ρ) if and only if for

1

2

every k ∈ N the sequence (fn(k))n∈N converges to f(k) in(Xk, dk).

(b) Assume that each of the spaces (Xk, dk) is separable. Provethat (X, ρ) is separable.

(c) Assume that each of the spaces (Xk, dk) is compact. Provethat (X, ρ) is compact. [Hint: We showed that a met-ric space is compact if and only if it is sequentially com-pact. Given a sequence (fn) in (X, ρ), show by inductionon k ∈ N how to find a subsequence (f

n(k)i

)i∈N such that the

following holds: (i) for each k ∈ N the sequence (fn(k+1)i

)i∈N

is a subsequence of (fn(k)i

)i∈N, and (ii) for each k ∈ N the

sequence (fn(k)i

(k))i∈N converges in (Xk, dk). Once this is

done, use part (a) to show that the diagonal subsequence(fn(i)i

)i∈N of (fn)n∈N converges in (X, ρ).]

(4) Let (X, dX) be a metric space. Let A be a compact subset of Xand let B be a closed subset of X. Assume that A and B aredisjoint. Prove that there exist disjoint open subsets U and Vof X, such that A ⊆ U and B ⊆ V .

(5) Let (X, dX) and (Y, dY ) be metric spaces and let f : X → Y bea continuous bijection. Prove that if (X, dX) is compact, thenf is a homeomorphism.

The Hilbert cube is the metric space ([0, 1]N, ρ), where ρ is the metricdefined by

ρ(a, b) :=∞∑k=1

2−k|ak − bk|

for all a = (a1, a2, . . . ) and b = (b1, b2, . . . ) ∈ [0, 1]N. By problem (3.c),([0, 1]N, ρ) is a compact metric space.

(6) Let (X, dX) be a metric space of diameter at most 1. Assumethat (X, dX) is separable and let (xk)k∈N be a countable densesubset of (X, dX). Define the map ϕ : X → [0, 1]N, by

ϕ(x) =(dX(x, x1), dX(x, x2), dX(x, x3), . . .

),

i.e., ϕ(x) ∈ [0, 1]N is the sequence ϕ(x)k := dX(x, xk).(a) Prove that the map ϕ : X → ϕ(X) is a homeomorphism

from (X, dX) to (ϕ(X), ρ).

3

(b) Use part (a) and problem (1) to conclude that every sep-arable metric space is homeomorphic to a subset of theHilbert cube.

Additional problems for honors section:

(7) Let c = (c1, c2, · · · ) ∈ RN and let

Ac = {x ∈ `2(N) : |xn| ≤ |cn| for all n ∈ N}.Prove that Ac is a compact subset of (`2(N), d‖·‖2) if and onlyif c ∈ `2(N).

(8) Let (X, d) be a compact metric space and let U be an open coverof X. Prove that there exists some δ > 0 with the followingproperty: for every x ∈ X there exists some U ∈ U such thatBδ(x) ⊆ U .

(9) Let (X, dX) be a metric space of diameter at most 1. LetK(X) denote the collection of all nonempty compact subsetsof (X, dX). Define the function dK(X) on K(X)×K(X) by by

dK(X)(K,L) = max{

supx∈K

dX(x, L), supy∈L

dX(K, y)}.

(a) Prove that dK(X) defines a metric on K(X).(b) Suppose that (X, dX) is separable. Prove that (K(X), dK(X))

is separable.(c) Let (Z, dZ) be another metric space of diameter at most

1, and let f : X → Z be a continuous map. Define f̃ :K(X)→ K(Z) to be the map f̃(K) := f(K), i.e., f̃(K) =

{f(x) : x ∈ K} for K ⊆ X compact. Prove that f̃ iscontinuous from (K(X), dK(X)) to (K(Z), dK(Z)).