# Show that all open sets in $\mathbb{R}^n$ are the same for metrics $d_1,d_2,d_\infty$

So I have shown that any $$x,y\in\mathbb{R}^n$$ that $$d_\infty(x,y) \le d_2(x,y)\le d_1(x,y)\le nd_\infty(x,y)$$, where $$d_\infty=\max\limits_{j\in\{1,\dots,n\}}\lvert x_j-y_j\rvert,$$ $$d_2(x,y)=\sqrt{(x_1-y_1)^2+(x_2-y_2)^2+\cdots+(x_n-y_n)^2},$$ $$d_1(x,y)=|x_1-y_1|+|x_2-y_2|+\cdots+|x_n-y_n|.$$ Will refer to this result as (1).

I have also shown that for metrics that satisfy $$d(x_1,x_2)\le kd'(x_1,x_2)$$ for some positive constant k and $$x_1,x_2 \in X$$ that any subset of $$X$$ that is $$d$$-open is also $$d'$$-open. Will refer to this result as (2).

I now have to show that the open sets in $$\mathbb{R}^n$$ are the same for the metrics $$d_1,d_2,d_\infty$$.

What I am wondering is that if the following answer is good enough: Let $$(\tau,d)$$ denote the family of all $$d$$-open subsets of $$X=\mathbb{R}^n$$. From (2) we have that $$(\tau,d_1)=(\tau,d_\infty)$$. From (1) we have that $$d_\infty(x,y)\le d_2(x,y)\le d_1(x,y)$$, so that must mean that $$(\tau,d_1)=(\tau,d_\infty)=(\tau,d_2)$$. So all open sets in $$\mathbb{R}^n$$ are the same for the three metrics.

From (2) we have that $$(\tau,d_1)=(\tau,d_\infty)$$.
But (2), as stated, starting with the hypothesis $$d \le k d'$$, does not let you conclude that $$(\tau,d)=(\tau,d')$$, it only says that $$(\tau,d) \subset (\tau,d')$$ because that's what it means to say "any subset of $$X$$ that is $$d$$ open is also $$d'$$ open".
So (2), when coupled with $$d_1 \le n d_\infty$$, only gives you the inclusion $$(\tau,d_1) \subset (\tau,d_\infty)$$ Similarly, (2) when coupled with $$d_\infty \le d_2$$, gives you the inclusion $$(\tau,d_\infty) \subset (\tau,d_2)$$ Perhaps you can see where this is going now, and can complete the proof.
• Ahh, so (1) and (2) combined gives that $(\tau,d_1)⊂(\tau,d_\infty)$ and $(\tau,d_\infty)⊂(\tau,d_2)$ and $(\tau,d_2)⊂(\tau,d_1)$, so that means that $(\tau,d_1)=(\tau,d_\infty)=(\tau,d_2)$. Is that correct then? – Nikolaj Nov 9 '18 at 11:23