What is the procedure to prove that a particular given metric generates a particular given topology?
Actually, I am given with a topology $\rho$ and a distance expression $d$ in my homework. Question is
Show that the topology $\rho$ is metrizable by showing that the distance $d$ defines a metric on $\mathcal T$ that generates the topology $\rho$.
I proved that the given expression $d$ is metric on space. But how to proceed next? How to show that the metric generates the topology?
You have to show two things: every $B_\rho(x,r)$ is open in $\mathcal{T}$ (this shows $\mathcal{T}_\rho \subseteq \mathcal{T}$) and vice versa, for every $O \in \mathcal{T}$ and every $x \in O$ there is some $r>0$ such that $B_\rho(x,r) \subseteq O$, which shows that $\mathcal{T} \subseteq \mathcal{T}_\rho$.
An example where I do this is here e.g.
The topology given by the metric is generated by the balls $$ B_r(x)=\{y:\ d(x,y)<r\}. $$ This means that every open set is a union of balls. So you have to show that every ball $B_r(x)$ is in $\rho$ , and that for every $V\in\rho$ and every $x\in V$, there exists $r>0$ such that $B_r(x)\subset V$.
