As discussed in an old post, in my module G12MAN Mathematical Analysis I introduced the notion of non-interior (nint) points for subsets of finite-dimensional Euclidean space in order to try to help students see clearly when a set was not-open by looking for the nint points.
In fact, I defined , so that
is open if and only if
is not open if and only if .
I wanted a suitable name for points in this set which I now denote by , and I originally called them non-interior points of . However students could think this meant points of . So my colleague Jim Langley suggested that it would be better to call them non-interior points in . (I suppose that one could also call them boundary points in .)
When my students sketch a subset of or of , they can now usually do a good job of indicating which boundary curves/points are included/excluded from the set, and accordingly they are able to see which points of the set are interior points and which points of the set are non-interior points. This helps them to see clearly whether a set is open or not. So far, so good.
However, in my recent exam, many students claimed (incorrectly!) that a union of non-open sets must be non-open. The argument they gave was that there must be lots of non-interior points in the final set, because there was at least one nint point in each of the sets making up the union. Somehow the ‘non-interiorness’ of the points has become separated from the set that they are non-interior for.
So, should the whole nint project be abandoned? Or should I just point out this problem and fix it? Or do I need to firmly attach the name of the set to the nint notation so that this confusion does not arise: I could have points and maybe also points for the two types of points in . But I still need a suitable name for the nint points, and preferably one that doesn’t cause confusion.