Question
$(X,\,d)$ is a metric space.
$f:\mathbb{R}_{\geq0} \longrightarrow \mathbb{R}_{\geq0}$ is a function such that:
- $f(x)=0 \Longleftrightarrow x=0$,
- $f$ is increasing,
- $f(x+y)\leq f(x)+f(y)\;\forall x,y\in\mathbb{R}_{\geq0}$.
$d_f=f\circ d$ is a distance on $X$.
$T_d$ and $T_{d_f}$ are the topologies generated by $d$ and $d_f$.
Give a metric space $(X,d)$ and a function $f$ such that $T_d$ and $T_{d_f}$ are not the same.
Attempt
Take $X=\mathbb{R}$ and $d$ to be the discrete distance.
Therefore $T_d$ is the discrete topology.
I am stuck here as I cannot find a function $f$ that gives a distance $d_f$ that generates a topology $T_{d_f}$ distinct from $T_d$.
More so, if I take $d$ as the usual distance, I am having trouble finding what $T_d$ and $T_{d_f}$ actually are.
Maybe I am not seeing something or I am overcomplicating it, because I feel intuitively that there might be a simple solution.
Any insight would be helpful.
Thank you.
Let $d$ be the usual distance in $\mathbb R$ and define $f(x)$ as $0$ if $x=0$ and $1$ otherwise. The new distance will be the discrete distance.