Collection vs. set of subsets vs. set of all subsets?
These get mixed. But the idea that I have is:
collection: a set of subsets, where order is ignored. So it's like a weaker form of set.
set of subsets: could be a collection, however since this is set of subsets, then one may not know whether the weakened properties of collections suffice. Or whether one actually has set of collections.
set of all subsets or the power set: this could be a collection, but it's not required to be one, because one could refrain from using the term collection and rather use subset.
family of sets or family of subsets is by definition a collection $F$ of subsets of a given set $S$. So it seems like here collection is the same as set of subsets, but also family of subsets. Or also, family is a set.
I wonder what sense does it make to use the terms collection and family at all? If they are sets, then why not call them sets?