In Bounded subset: Only in metric space or also for premetric space? it was explained that - in general - a premetric space $(X, m)$, where \begin{align} x,x'&\in X\\ m(x,x')&\ge0 \\ m(x,x)&=0 \\ m(x,x')&=m(x',x) \end{align}
cannot be a Bornological space. In other words, a subset $X'$ of the underlying set $X$ of the premetric space can only be bounded if the "premetric" is actually a metric (additionally fulfilling the triangular inequality).
Now I would like to ask if there is a way to have my premetric not fulfill the triangular inequality, but still enable bound-ability. What alternative condition could I impose on the premetric, which is more general than the triangular inequality but still allows to talk about bounded subsets of $X$.
Here is my suggestion for the subset $X'\subset X$:
$$ \exists r > 0: r\in\mathbb R : m(x,x') < r, m(x,x'') > 0 \;\forall\; x,x' \in X', x''\in X\setminus X' $$
