414/514 – Metric spaces

This is homework 2, due Monday September 26 at the beginning of lecture.

Let (X,d) be a metric space.

  • Show that if d_1:X\times X\to{\mathbb R} is defined by \displaystyle d_1(x,y)=\frac{d(x,y)}{1+d(x,y)}, then d_1 is also a metric on X.
  • Show that if U is open in (X,d) then it is open in (X,d_1), and viceversa.

Recall that U is open iff it is a union of open balls. Use this to explain why it suffices to show that if U is open in (X,d) then for any x\in U there is an \epsilon>0 such that

B_\epsilon^{d_1}(x):=\{y\mid d_1(x,y)<\epsilon\}\subseteq U,

and similarly, if V is open in (X,d_1) then for any z\in V there is a \delta>0 such that

B_\delta^d(z):=\{w\mid d(z,w)<\delta\}\subseteq V.

In turn, explain why to show this it suffices to prove that for any x\in X and any \eta>0 there is a \rho>0 such that

B^{d_1}_\eta(x)\supseteq B^d_\rho(x)

and, similarly, for any \tau>0 there is a \mu>0 such that

B_\tau^d(x)\supseteq B^{d_1}_\mu(x).

Finally, prove this by showing that we can take \rho=\epsilon (no matter what x is) and similarly, find an appropriate \mu that works for \tau (again, independently of x).

  • Illustrate the above in {\mathbb R}^2 as accurately as possible.
  • Suppose that a sequence (x_n)_{n\in{\mathbb N}} converges to x in (X,d) and to x' in (X,d_1). Show that x=x'.
  • Is it true that a sequence (x_n)_{n\in{\mathbb N}} is Cauchy in (X,d) iff it is Cauchy in (X,d_1)? (Give a proof or else exhibit a counterexample, with a proof that it is indeed a counterexample.)
  • (*) Show that any dense G_\delta subset of {\mathbb R} has the same size as {\mathbb R}.

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