Theorem20.1. Let $X$ be a metric space with metric $d$. Defined $\bar d-:X×X→R\ $ by the equation $\bar d ( x , y ) = \min \{ d ( x , y ) , 1 \}$ . Then $\bar d$ is a metric that induces the same topology as $d$.
Does it mean that $\bar d$ "can" induce the same topology as $d$ or always induces the same topology as $d$?
or for the given metric, there always exists unique topology?
It is true that a metric always induces a topology, but it is not true that the topology is unique to that metric, i.e., different metrics may induce the same topology.
For the sake of making this more clear, let's make the following definition: For a metric $d$ on $X$, we say a subset $U$ of $X$ is $d$-open if for each $x_0\in U$, there is some $\varepsilon>0$ such that if $x\in X$ and $d(x,x_0)<\varepsilon$, then $x\in U$. The collection of $d$-open subsets is then called the topology induced by $d$.
Thus, when we say that $\overline d$ induces the same topology as $d$, we mean that the collection of $\overline d$-open subsets of $X$ is precisely the collection of $d$-open subsets.