Q:

What does induced (by a metric) topology mean? 

I know that this is a question that has already been discussed here, but none of them provides me an answer that I understand, unfortunately. 
I know the properties that should be satisfied by a metric, a topology and what a ball is; however, I fail to establish the link between metric spaces and topological spaces.  My main problem is, in fact, I do not understand what is meant by metrizable topology. 

A:

Let $X$ be a set and let $\tau$ be a topology in $X$. We say that the topological space $(X,\tau)$ is metrizable if there is a distance $d\colon X\times X\longrightarrow\mathbb R$ such that the elements of $\tau$ are the open sets of the metric space $(X,d)$. In other words, if$$\tau=\left\{A\subset X\,\middle|\,(\forall x\in A)(\exists r>0):B_r(x)\subset A\right\},$$where $B_r(x)=\{y\in X\,|\,d(x,y)<r\}$.

