Find the interior and boundary of `[0,2]nn[2,4]`

Expert Answers
degeneratecircle eNotes educator| Certified Educator

Note that `[0,2]nn[2,4]={2},` a set with just one element. No neighborhood of 2 (in this case a neighborhood is an open interval of the number line centered at 2) can possibly be contained in the set `{2}`, so there are no interior points and thus the interior of the set is empty.

The boundary of the set can be defined in several equivalent ways. One definition for the boundary of a set `S`, if `Ssubset RR`, is the set of all points `x` in `RR` such that every neighborhood of `x` contains at least one point in `S` and one point not in `S.` Thus the boundary of `{2}` is `{2}` itself. There is another definition involving closures as well.

To reiterate, the interior of `[0,2]nn[2,4]` equals `O/` and the boundary of `[0,2]nn[2,4]` equals `[0,2]nn[2,4]` , or simply `{2}.`