The setting is as follows. Suppose $(X,\mathrm{d})$ is a metric space. For any $(x,r)\in X\times \mathbb {R}_+$, denote $\mathrm{B}(x,r)$ as the closed ball in $X$, i.e. $\mathrm{B}(x,r)=\{y\in X\left|\ \mathrm{d}(y,x)\le r\right.\}$. We know that for any $(x_1,r_1)$ and $(x_2,r_2)$ satisfying $\mathrm{d}(x_1,x_2)\le r_2-r_1$, by the triangle inequality, we have $\mathrm{B}(x_1,r_1)\subset \mathrm{B}(x_2,r_2)$. This fact holds for all metric spaces.
When we think about the converse of this fact, i.e. "For any $(x_1,r_1)$ and $(x_2,r_2)$ satisfying $\mathrm{B}(x_1,r_1)\subset \mathrm{B}(x_2,r_2)$, we have $\mathrm d(x_1,x_2)\le r_2-r_1$." It turns out to be not always true for any metric space $(X,\mathrm{d})$. So the question is
Question. How can we give some nice conditions to distinguish the metric spaces where the property $\mathrm{d}(x_1,x_2)\le r_2-r_1$ and $\mathrm{B}(x_1,r_1)\subset \mathrm{B}(x_2,r_2)$ are equivalent? (Perhaps, you need to change the way we ask this question to make it more reasonable.)
Denote the set of such metric spaces as $\mathcal {E}$. We have known that if $\mathrm{diam}(X)<\infty$, we have $X\notin\mathcal {E}$, since the radius of the ball can be chosen to be larger than the diameter of $X$. Thus, if $X\in\mathcal {E}$, $X$ must be unbounded, i.e. $\mathrm{diam}(X)=\infty$.
We have also known that every Euclidean space of finite dimension with the usual metric belongs to $\mathcal {E}$. Every dense subset of a Euclidean space of finite dimension as a metric subspace also belongs to $\mathcal {E}$. For any $n\in\mathbb N=\mathbb{Z}_{\ge 0}$, the upper-half space $\mathbb{R}^n\times\mathbb{R}_{\ge 0}$ as a subspace of $\mathbb {R}^{n+1}$, however, doesn't belong to $\mathcal {E}$ (please see the following picture).
Hence, we guess if $X\in\mathcal {E}$, $X$ should possess some completeness in some sence, just like the Euclidean space of finite dimension.
Moreover, we can prove that every parabola on $\mathbb {R}^2$ doesn't belong to $\mathcal {E}$, since the parabola is pointed or sharp at the extreme point, particularly when we enlarge the parabola (please see the following picture).
Meanwhile, we can prove that every sin function on $\mathbb {R}^2$ doesn't belong to $\mathcal {E}$ (please see the following picture (sorry; my vpn doesn't work suddenly)).
Finally, maybe we can prove that every unbounded $C^\infty$ plane curve which is self-intersection point free and not a straight line or a curve with precisely one endpoint doesn't belong to $\mathcal {E}$.
Hence, we also guess if $X\in\mathcal {E}$, $X$ may possess some flatness in some sense, just like the Euclidean space of finite dimension. We guess this question may not only be a point set topology question, but also need some differential topology.
I cannot find any material about this question on the Internet. If you can give me some references or tell me some advanced topics, I will be more than grateful. Thank you for bearing my unreasonable question.
One sufficient condition is:
$\exists (y_n)_{n \in \mathbb N},$ when $n \to +\infty, d(x_1,y_n) \to r_1^-$ (meaning $d(x_1,y_n) \le r_1$ and $d(x_1,y_n) \to r_1$) and $d(x_2, y_n) \to r_1 + d(x_2,x_1)$.
Whether this is a nice condition is subject to appreciation :-)
Proof: suppose we have $B(x_1,r_1) \subset B(x_2,r_2)$ and
$\exists (y_n)_{n \in \mathbb N},$ when $n \to +\infty, d(x_1,y_n) \to r_1^-$ and $d(x_2, y_n) \to r_1 + d(x_2,x_1)$.
Then $d(x_2, y_n) \le r_2$, so
$d(x_2,x_1) = \lim_{n \to \infty} (d(x_2, y_n) - r_1) \le \lim_{n \to \infty} (r_2 - r_1) = r_2-r_1$.