How can I generally show that two metric spaces induce the same topology, for instance, in my textbook it says metric space $d(x,y$) and $\sqrt{d(x,y)}$ induce the same topology. How do I show this? Much help would be appreciated.
ok so very generally I wrote up some solution, and it goes:
Let $d^{'}=\sqrt{d}$, then we want to show that $B_d(x,\epsilon) \subseteq B_{\sqrt{d}}(x,\epsilon)$. Let $y \in B_d(x,\epsilon)$,but $d^{'}$ is the square root metric of $d$, so we have $d^{'} \le d$, hence $d^{'}(x,y)\le d(x,y)<\epsilon \implies y \in B_{\sqrt{d}}(x,\epsilon)$, therefore $B_d(x,\epsilon) \subseteq B_{\sqrt{d}}(x,\epsilon)$. is this the correct way?