This is an exercise problem from Munkres's Topology (Exercise 11 of Section 20 "The Metric Topology", 2nd edition).
Exercise 11: Show that if $d$ is a metric for $X$, then $$d'(x,y) = d(x,y) / (1 + d(x,y))$$ is a bounded metric that gives the topology of $X$.
It is not hard to show that $d'$ is indeed a bounded metric.
For the rest (i.e., $d'$ induces the topology $X$), I found at this site [dbfin] a proof using the following lemma (which is given as the exercise 3 of same section).
Lemma: Let $X$ be a metric space with metric $d$. Let $X'$ denote a space having the same underlying set as $X$. If $d: X' \times X' \to \mathbb{R}$ is continuous, then the topology of $X'$ is finer than the topology of $X$. (Note: its proof can be found here; more elaborations can be found at Metric space and continuous function.)
The proof I found is quite brief:
Solution from [dbfin]: Since the functions $f: f(x) = x / (1 + x)$ and $f^{-1}$ are continuous, $d'$ is continuous in $d$ and vice versa, which means the topologies are the same.
I am quite confused about the condition of the Lemma (if $d: X' \times X' \to \mathbb{R}$ is continuous) and therefore don't understand the details of the proof.
My Problem: In order to show that the two topological spaces $X$ (induced by $d$) and $X’$ (induced by $d’$) are the same. We can show that (1) $X’$ is finer than $X$ and (2) $X$ is finer than $X’$.
To prove the former one (the latter one goes similarly), using the lemma above, we shall first justify its premise (i.e., the “if ” part) $d : X’ \times X’ \to \mathbb{R}$ is continuous. I don’t know how this is achieved in the [dbfin] solution. Could someone offer me more details?
P.S.: The "more standard" approach without the lemma above is given by @Stromael. However, I have found a similar proof at this site after I posted this problem. To avoid duplication, in this post, I would prefer to the solution with the lemma.
Best Answer
Hint: a composition of continuous functions is continuous, and two topologies are equal if and only if they are one another's refinements.