To what other structures does this generalize?
The intersection of all subgroups containing a subset $ $ $S$ $ $ is $ $ $<S>$ $ $.
The intersection of all ring ideals containing a subset $ $ $S$ $ $ is $ $ $<S>$ $ $.
The intersection of all topologies containing a basis $ $ $B$ $ $ is $ $ $<B>$ $ $.
The intersection of all vector spaces containing a subset $ $ $S$ $ $ is $ $ $<S>$ $ $
And so on. All of these are the smallest structure containing the other. I have seen this pattern popping up in my classes a lot recently, but bear in mind I have NOT taken universal algebra or anything of the sort, only vaguely read some things. It reminds of Noether's 4 isomorphism theorems, which generalize to many algebraic structures. I suppose this generalizes to any other type of structure in which the intersection is also one such stucture.
