I had thought that subcollection referred to a collection of sets in some topological space $(X,\tau)$, where each set is a subset of $X$
But I also see it being used in subbasis, and subspace topology so I have gotten a little bit unsure.
What is the proper definition of subcollection and its usage?
Is it collection to say: "Let $\{u_\alpha\}_{\alpha \in I}$ be a subcollection of sets in $\tau$"?
"Collection" and "subcollection" are just synonyms of "set" and "subset" that are sometimes used to avoid confusion (particularly when the elements of these sets are themselves sets). So a collection $A$ of sets is just a set of sets, and then a subcollection of $A$ is just a subset of $A$ (i.e., a set of sets, all of which are elements of $A$). The term "subcollection" on its own has no meaning; it is always used to refer to a subcollection of some particular other collection.