## 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 ${\mathbb R}$.