This is a straightforward generalization of the customary definition of totally bounded sets on uniform spaces:
Definition Reloid is totally bounded iff for every there exists a finite cover of such that .
See here for definitions and notation.
I don’t know which interesting properties totally bounded spaces have (except of their connection to compact spaces, but at the time of writing this compactness of funcoids is not yet properly defined).
Please post comments about properties of totally bounded spaces, in order to develop the theory further.