I saw the notion "small disk" very frequently used in literature. For example, in Brunnian braids on surfaces by V. G. Bardakov, R. Mikhailov, V. V. Vershinin, J. Wu, one line reads:
Let $P_n(M)$ be the $n$–strand pure braid group on $M$. Let $D^2$ be a small disk in $M$.
(Here $M$ is a compact connected surface.)
I am wondering what on earth is a "small" disk? (Topologically) a disk is a disk, smallness does not make sense to me if there is no metric defined. Even in a metric space, how small is small?
If you actually look at the paper, there is no content to the word small here. It is purely for flavor, i.e. a suggestion to the reader on how to visualize this disk.
There are cases in which a statement of this sort is shorthand for something both meaningful and precise, such as "for all $x \in M$ there exists a neighborhood $U$ of $x$ such that for all disks $D \subset U$ containing $x$ $\ \ldots$" (exactly what statement is intended is hopefully clear from context in these cases).