Noun
subcollection (plural subcollections)
A subset of a collection.
Closed sets also give a useful characterization of compactness: a topological space X is compact if and only if every collection of nonempty closed subsets of X with empty intersection admits a finite subcollection with empty intersection. Source: Internet
The first one describes that one item is an instance of some collection, the second one that one collection is a subcollection of another one. Source: Internet