Continuing on the theme of quantifier packaging …
Currently I teach second-year mathematical analysis at Nottingham, and this includes the topology of finite-dimensional Euclidean space.
As the students were much happier with the notion of an interior point of a set than they were with the notion of an open set (one less quantifier), I decided to package this as follows. Every set S is the disjoint union of int(S), the set of interior points of S (standard), and nint(S), the set of non-interior points in S (non-standard).
With this terminology, the set S is open if and only if S = int(S) (or, equivalently, if every point of S is an interior point of S). In terms of nint(S), S is open if and only if nint(S) is the empty set, i.e., if and only if there are no non-interior points in S. (This makes it particularly easy to see that the empty set is open!)
The set S is not open if and only if nint(S) is non-empty, i.e., if and only if there is at least one non-interior point in S.
It is very easy for students to study lots of examples of sets S and to investigate whether particular points of the set are in int(S) or nint(S). From what I have seen, the students get on with this very well, and they can demonstrate that sets are not open by pointing out that certain points are clearly non-interior points.
Thanks are due to my colleague Jim Langley for the following: I originally used the terminology ‘non-interior point of S’, but this was ambiguous and some students assumed that points of the complement of S would count. Jim suggested I use ‘non-interior point in S’ instead.
Note that, with this terminology, the boundary of S is the disjoint union of nint(S) and nint(Sc).