Imagine that we have a collection of sets $\cal C$, let $S\notin \cal C$ and $A \in \cal C$.
In algebra, topology, etc... we often see statements like " ----- is the smallest set containing -----", but I'm always not sure how to formalize these kinds of statements. How can I write that $A$ is the smallest set in $\cal C$ containing the set $S$? How can we measure the "length" of these sets in order to say that one is smaller than the other?