Let $(X,d_1)$ and $(X,d_2)$ be two metric spaces which have the same infinite set $X$, but the different metrics $d_1$ and $d_2$.
Denote the collection of subsets $X$ by $S$, and the collection of all open subsets of $(X,d)$ by $U_{d}$.
Then, is it possible to know $|U_{d_1}|$ and $|U_{d_2}|$, the cardinalities of the collections of all open subsets of $(X,d_1)$ and $(X,d_2)$?
For example, if $X=\mathbb{R}$ and if the metric $d$ on $\mathbb{R}$ is defined to be the discrete metric, then $|U_d|=|S|.$
But, if $X=\mathbb{R}$ and if the distance function $d$ of $X$ is defined to be the usual Euclidean distance function, $|U_d|=c$.
In conculusion, what I'm asking is that is it possible to characterize $|U_d|$ of a metric space $(X,d)$, with $X$ being infinite, in terms of metric function $d$?
Also, is it possible to relate $|S|$ with $|U_d|$ of a metric space $(X,d)$ in terms of metric function $d$?
Let $X$ be an infinite set.
Note that a topology $\tau$ on X is a collection of subsets of $X$, that is $\tau\subset P(X)$.
We consider the collection $T$ of all topologies on X.
Pick a topology $\tau$ in $T$ one at a time, and let $(X,\tau)$ be the corresponding topological space.
If $(X,\tau)$ is a metrizable space, then let a corresponding metric space be $(X,d)$. Then, $|U_d|$ of the metric space $(X,d)$ is just equal to $|\tau|$. Note that there may be more than one metrics $d$ such that $(X,d)$ is a corresponding metric space to $(X,\tau)$.
If $(X,\tau)$ is not a metrizable space, then there is no metric $d$ such that $\tau$ is the induced topology by $d$, and hence for any metric $d$ on $X$ we will not have $\tau$ as a collection of open sets of a metric space $(X,d)$.
This gives an abstract algorithm to obtain the possible cardinality of a collection of all open sets of a metric space $(X,d)$ from an infinite set $X$.
Lastly, note that $|U_d|\leq |P(X)|$ for any metric $d$ on $X$.