"Is in the neighborhood of"-style relations

Question

Consider a nonempty set $S$ with binary relations $\preccurlyeq$ and $\sim$ defined on it such that:

1. $(S, \preccurlyeq)$ is a total ordering.
2. $(S, \sim)$ is reflexive and symmetric.
3. For all $x,y,z\in S$, if $x\preccurlyeq y\preccurlyeq z$ and $x\sim z$, then $x\sim y$ and $y\sim z$.

Examples of such $x\sim y$ on $\mathbb{R}$ include $|x-y|\leq 5$ and $\lfloor x\rfloor = \lfloor y\rfloor$.

Does either the relation $\sim$ or property 3 have a name?