Topological spaces are equipped with a bare minimum of structure to allow for a formalization of the statement "the net $a$ converges to the point $x$." Actually this isn't strictly true, but its true enough for the purposes of this question.
Now supposing we have a metrical structure, then we can formulate many more concepts. In particular, if $X$ is a metric space and $a$ and $b$ are sequences in $X$, then we can define that "$a$ and $b$ converge toward each other" iff
$$\lim_{i \rightarrow \infty}d(a_i,b_i)=0.$$
This defines an equivalence relation on sequences in $X$. Unfortunately, no similar equivalence relation can be defined in the presence of a mere topology.
Question. What do we call those abstract structures that have the bare minimum of structure to support a definition of "nets that converge to each other," and how are these structures defined?