We will use the following notion of equivalence of metrics. Let $X$ be a set, and let $d_1$, $d_2$ be two distance functions on $X$, such that $(X, d_1)$ and $(X,d_2)$ are both metric spaces. We say $d_1$ and $d_2$ are equivalent if there exists positive constants $c_1, c_2$, such that for any $p,q \in X$, we have $$ c_1 d_1(p,q) \leq d_2(p,q) \leq c_2 d_1(p,q). $$

1. Equivalence of metrics on $\R^2$. Recall that we can equip $\R^2$ with the usual Euclidean metric $d_2(\vec x, \vec y) = \sqrt{|x_1 - y_1|^2 + |x_2- y_2|^2}$, or with the metric $d_\infty(\vec x, \vec y) = \max(|x_1 - y_1|, |x_2 - y_2|)$. Prove that $d_2$ and $d_\infty$ are equivalent.

2. Equivalent metrics induces the same topology. Let the set $X$ be equipped with two equivalent metrics $d_1, d_2$. Consider two types of balls $$ B_r^{d_i}(p) = \{ p' \in X \mid d_i(p,p') < r\} $$

• prove that for any $p \in X$, $r > 0$, and any $q \in B_r^{d_1}(p)$, there exists $\epsilon>0$, such that $B_\epsilon^{d_2}(q) \subset B_r^{d_1}(p)$.
• prove that if a subset $U \subset X$ is open with the $d_1$-metric, then it is open in the $d_2$-metric.

3. Addition on $\R^2$ is continuous. Let $f:\R^2 \to \R$ be given by $f(x,y) = x+y$. Prove that $f$ is continuous. You can use either one of the following definition for continuous functions

• (metric space sense) $f: (X, d_X) \to (Y, d_Y)$ is continuous if for any $x \in X$, and any $\epsilon>0$, there exists $\delta>0$, such that $f (B_\delta(x)) \subset B_\epsilon(f(x))$.
• (topological sense) $f: X \to Y$ is open, if for any open subset $V$ in $Y$, $f^{-1}(V)$ is open.

4. Let $X$ be a metric space $A \subset X$ be any subset. Define the distance function to $A$ as $$ d_A(p) = \inf \{ d(p,q) \mid q \in A \} $$ Prove that $d_A$ is a continuous function on $X$, and show that $d_A(p)=0$ if and only if $p \in \bar A$.

5. Let $P = \{\vec a = ( a_1, a_2, \cdots, ) \mid a_i \in \R, a_N=0 \text{ for } N \gg 0 \}$ be the set of real valued sequences, such that each sequence only has finitely many non-zero entries. Consider the metric functions $d_2(\vec a, \vec b) = \sqrt{\sum_{n=1}^\infty |a_n - b_n|^2}$, and $d_\infty(\vec a, \vec b) = \max\{|a_n - b_n|, n=1,2,\cdots \}$. Prove that the two metric defines different topology on $P$. Hint: prove that there does not exists $c_1, c_2>0$, such that for all $\vec a \in P$, we have $$ c_1 d_2(0, \vec a) < d_\infty(0, \vec a) < c_2 d_2 (0, \vec a). $$