There are
Lemma interior_subset {T} [A : set T] : A° `<=` A
Lemma subset_closure {T} [A : set T] : A `<=` closure A
Lemma closure_subset [T] [A B : set T] : A `<=` B -> closure A `<=` closure B
It looks like the first two are named according to the position of each operator wrt the subset relation.
The third one looks ad-hoc, and especially problematic if we want to add its interior counterpart.
Lemma subset_interior [T] [A B : set T] : A `<=` B -> interior A `<=` interior B
This name would be bad.