Can I think of a toy contour as any closed piecewise-smooth curve which is simple?

Question

I feel Stein doesn't give the specific definition of toy contour, can I think of a toy contour as any closed piecewise-smooth curve which is simple?

Answer 1

Yes, that's a valid way to think about it.

What's going on here is a theorem of topology:

The Jordan Curve Theorem. If $C$ is any continuous simple closed curve then $\mathbb R^2 - C$ has two connected components, one of which (called the "interior") is bounded and the other (called the "exterior") is unbounded.

It's pretty clear from what the author is saying (in the first paragraph of the passages you have shown us) that he/she does not want to spend the time to either state or prove the Jordan Curve Theorem. This is certainly a reasonable decision for a textbook author, because it is a very hard theorem to prove.

Instead, the words "toy contour" are used to mean any continuous simple closed curve for which the conclusion of the Jordan Curve Theorem is obvious. Examples of this would include any convex simple closed curve (an exercise). But he also wants some nonconvex examples such as the "keyhole" example, and the exercise is still pretty easy for those.

And, by the way, there is another theorem with the same hypotheses but still stronger conclusions than the Jordan Curve Theorem, it is known as the Schönflies Theorem. The author's comments in that first paragraph lead me to think that the concept of "toy contour" might also be referring to those stronger conclusions in some form or another.