We know that $W_2(f,z) = \#F^{-1}(z)$ mod $2$. That is, $f$ winds $X$ around $z$ as often as $F$ hits $z$, mod $2$, where $\partial F = f$.
Then the text says $W_2(X,z)$ must be $1$ or $0$, depending whether $z$ lines inside or outside of $X$. To me, this sounds like
- $W_2(X,z)$ must be $1$ if $z$ lies inside of $X$.
- $W_2(X,z)$ must be $0$ if $z$ lies outside of $X$.
However, I thought
- $W_2(X,z)$ must be $1$ if $f$ wraps around $z$ for odd times.
- $W_2(X,z)$ must be $0$ if $z$ wraps around $z$ for even times, or lies outside of $X$.