The closed sets in the plane with the GTPP appear at first glance to be similar to the closed sets with the ordinary TPP or the STPP. That is we expect the closed sets with the GTPP to be squares with square holes. Indeed, these sets certainly have the GTPP, but the class of infinite closed sets with the GTPP is a more interesting one than for the ordinary TPP or the STPP.

The key observation is that any monotone function f whose graph is in R², while it doesn't have the GTPP, has the property that any grid square cuts its graph into at most three pieces. We now see that the infinite closed region above (or below) f has the desired properties. In fact, we need not even require that this boundary curve be a function, since vertical lines work as well as horizontal lines.

Proposition 5. Let A be a curve that can be parametrized by monotone functions x = u(t) and y = v(t). Then the set of points above (or below) A including A itself has the GTPP.

Proposition 6. The closed sets in R² with the GTPP are precisely the grid squares, closed square regions with a countable number of open square regions removed, and regions bounded by a monotone function (as described above) with a countable number of open square regions removed.

The next section deals with some surfaces in R³.