Suppose $f(z)$ is analytic in an open region $D$ which is not necessarily simply connected, and $C$ is a closed rectifiable contour in $D$. Then $\oint_C f(z)dz = 0$
if either (a) $C$ is contained in a simply connected subregion of $D$, or (b) $C$ forms the oriented boundary of a simply connected subregion of $D$.
The part (a) is obvoiusly true by Cauchy integral theorem but I couldn't understand why (b) is true. How can we be sure that $C$ itself lies in simply connected subregion? I think it's a consequence of $D$ is an open region but I couldn't prove that.
Here is the source of the statement: http://www.maths.usyd.edu.au/u/olver/teaching/MATH3964/cauchy.pdf