Prove that a set is compact with respect to a different metric in $\mathbb{R}$ Prove that a set $A\subset \mathbb R$ is compact with respect to $d_1$ if and only if it is compact with respect to $d$. Were $d$ is the standard metric on $\mathbb{R}$ and $d_1(x,y)=\min(|x-y|,1)$. 
I know that $A$ is compact it its closed and bounded, I was thinking to prove that its sufficent to show that it its closed with respect to the other metric, because close sets of compact sets are compact. Any help I would appreciate. Maybe Im doing the wrong approach 
 A: HINT: Compactness is a topological property, not a metric property: the real definition of compactness is that $A$ is compact if and only if every cover of $A$ by open sets has a finite subcover, and this depends only on the topology (the collection of open sets), not on the metric used to define them. Thus, it suffices to show that $d_1$ and $d$ define the same open sets. 
Thus, you want to show that for each $U\subseteq\Bbb R$ the following two statements are equivalent:


*

*For each $x\in U$ there is an $\epsilon>0$ such that $B_d(x,\epsilon)\subseteq U$.

*For each $x\in U$ there is an $\epsilon>0$ such that $B_{d_1}(x,\epsilon)\subseteq U$.
This will follow if you can prove the following stronger statement:


*

*For any $A\subseteq\Bbb R$ and $x\in A$ there is an $\epsilon>0$ such that $B_d(x,\epsilon)\subseteq A$ if and only if there is an $\epsilon>0$ such that $B_{d_1}(x,\epsilon)\subseteq A$.


If you get stuck, take a look at the small further hint in the spoiler-protected block below.

 If there is an $\epsilon>0$ such that $B_d(x,\epsilon)\subseteq A$, then there is a positive $\epsilon<1$ such that $B_d(x,\epsilon)\subseteq A$. The same is true for the metric $d_1$.

A: Two metrics on a space $X$ are said to be equivalent when they induce the same topology on $X$. Show that whenever $d_1,d_2$ are metrics that satisfy
$$\exists\, c,C \in \mathbb{R}, \, \, \forall a,b \in X, \\
c\cdot d_1(a,b)\leq d_2(a,b) \leq C\cdot d_1(a,b)$$
they induce the same topology on $X$. What are $X, d_1$ and $d_2$ in your case? Can you find values of $c,C$ for them?

Hint: How can you verify a set is open in a metric space, using its metric?

