Let $X$ be an arbitrary set and to start with, let us consider a relation $\leq$ on $X$ (that is $\leq$ is a subset of $X^2$) which is reflexive and transitive. such a relation is called a preorder. Now, let us consider a similar object - a family of relations $\leq^\delta$ parametrized by $\delta\in[0,\infty)$ that satisfy the following properties:
$x\leq^0 x$ for any $x\in X$,
if $x\leq^\delta y$ and $y\leq^{\epsilon} z$ then $x\leq^{\delta+\epsilon} z$ for any $x,y,z\in X$.
The first condition is obviously reflexivity, whereas the second can be understood as "approximate transitivity". It reminds of the triangular inequality. In fact, the function $$ d(x,y):=\inf\{\delta>0:x\leq^\delta y \text{ and }y\leq^\delta x\} $$ can be shown to be a pseudo-metric (a metric that can be zero outside of the diagonal). Is there any special name for the family $\leq^\delta$, or for its symmetrization $\approx^\delta$ where $$ x\approx^\delta y \iff x\leq^\delta y \text{ and }y\leq^\delta x. $$ Let me also mention that $\leq^0$ alone is a pre-order.