Question on two metric spaces properties

Question

Question:

Let $X$ be a set and let $d_1$ and $d_2$ be two metrics on $X$. Assume that there exists a constant $C > 0$ such that
$d_1(x, y) \le C\, d_2(x, y)\ \ \ \forall x, y \in X$.

1. Show that if $E \subset X$ is an open set in the metric space $(X, d_1)$, then $E$ is also an open set in the metric space $(X, d_2)$.
2. Show that if $(X, d_2)$ is a compact metric space, then $(X, d_1)$ is also a compact metric space.

Answer 1

Hints:

1) Suppose $E$ is open in $(X,d_1)$, and let $x\in X$. If the distance $d_2$ of $y$ to $x$ is small, then the distance $d_1$ of $y$ to $x$ is also small. That way, every open ball in $d_1$ contains and open ball (with same center) in $d_2$. Then, just apply the definition of open sets.

2) If $\mathscr{U}$ is a open cover of $(X,d_1)$, then, by item 1, it is also an open cover for $(X,d_2)$. Just take a finite subcover.

Answer 2

Let's set $B_1(x,r)=\{y\in X:d_1(x,y)<r\}$ and $B_2(x,r)=\{y\in X:d_2(x,y)<r\}$

Consider the identity map $j\colon (X,d_2)\to (X,d_1)$. Prove that $j$ is continuous, which has the obvious consequence that every open set in $(X,d_1)$ is also open in $(X,d_2)$.

If $(X,d_2)$ is compact, also $(X,d_1)$ is compact, because the image of a compact set under a continuous map is compact.