Are single points or rectangles of area $0$ simple closed curves?

Question

I was solving the following problem:

Show that $\oint_C{-x^2ydx+xy^2dy} > 0$ around any simple closed curve $C$.

I began with applying Green's Theorem: $\oint_C{-x^2ydx+xy^2dy} = \iint_R{y^2+x^2}$

$y^2+x^2 \geq 0$, so $\iint_R{y^2+x^2} \geq 0$.

Note that I still need to prove that $...>0$, not $...\geq 0$.

The only time $... = 0$ is when $R$ has an area of $0$, which means it's a point or a rectangle with width or height of $0$.

So, if I can say that such $R$s aren't simple closed curves, than I can adjust $\geq$ to $>$.

Are single points or rectangles of area $0$ simple closed curves?

Answer 1

No. According to the Jordan curve theorem, all Jordan curves (also known as simple closed curves) divide the plane into two regions - an inside and an outside.

A single point has no "inside" in this sense.