Non-interior points revisited

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 $\mathbb{R}^n$ 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 $\text{nint}\,A = A \setminus \text{int}\,A$, so that

$A$ is open if and only if $\text{nint}\,A = \emptyset$

and

$A$ is not open if and only if $\text{nint}\,A \neq \emptyset$.

I wanted a suitable name for points in this set which I now denote by $\text{nint}\,A$, and I originally called them non-interior points of $A$. However students could think this meant points of $\mathbb{R}^n \setminus \text{int}\,A$. So my colleague Jim Langley suggested that it would be better to call them non-interior points in $A$. (I suppose that one could also call them boundary points in $A$.)

When my students sketch  a subset of $\mathbb{R}$ or of $\mathbb{R}^2$, 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 $A\text{-nint}$ points and maybe also $A\text{-int}$ points for the two types of points in $A$. But I still need a suitable name for the nint points, and preferably one that doesn’t cause confusion.

2 Responses to Non-interior points revisited

1. David

So the introduction of $nint(A)$ makes it much harder to make the mistake of starting a proof about openness with the line:
$\exists r>0$ such that $\forall x \in A, B_r(x)\subset A$?

Whereas with $nint(A)$ there is a tendency to make mistakes arising from thinking of a point $x$ as a non-interior point, without reference to the set $A$ itself? (I think this is what you’re observing)

I can see how it looks like this problem comes out of discussion of non-interior points individually rather than he idea of $nint(A)$ itself, which is a set. I guess the question is whether or not there are other possible mistakes lurking around the corner caused by thinking too much about non-interior points. If not, then a little lemma about $nint(A) \cup nint(B) \neq nint(A \cup B)$ might sort things out.

I’m not sure I see the two definitions of open:
(a) $A$ is open if around every point of $A$ a ball can be drawn etc.. etc..
(b) $A$ is open if all its points are interior points

as really very different since it seems like the hard work has just been moved in (b) into the definition of interior. From that point of view it may be unnatural to offer (b) without also offering (a) as an alternative formulation at the same time.

Overall the use of $nint(A)$ to hide one quantifier from the big definition is definitely a technique used usefully elsewhere like in limits of functions (through sequences) — but you do risk not exposing students to these more complex definitions with more than one quantifier in so that when they come across them down the track they don’t know how to grapple with them.

2. I always offer the alternative, full, formulations at the same time for comparison and parsing.
Moreover, I work through the negations of these definitions in both short and long forms too.
“Moving most of the work” in this case means looking at a simpler definition (with one fewer quantifiers) first, and then defining the big definition using the simpler concept once students have seen enough examples to be happy with the easy concept.
I think that doing function limits using sequences instead of epsilon-delta is probably a much more significant issue, and is one which I have discussed in another post. Because you can not make an easy step up from understanding the sequence definitions to understanding the epsilon-delta definitions.