TITLE: Why do we require the (whole) domain to be simply connected in Cauchy's theorem, residue theorem, etc.?
QUESTION [11 upvotes]: Main Question
A very common form of Cauchy's integral theorem is

Let $A\subset\mathbf{C}$ be simply connected and open, $f:A\rightarrow\mathbf{C}$ analytic, $I$ a compact interval, and $\gamma:I\rightarrow A$ continuous (or whatever) with $\gamma(I)\subset A$. Then $\int_\gamma f(z)dz=0$.

I think it would also suffice (and not make the proof any harder) to assume that $\gamma$ is homotopic in $A$ to a point without requiring that $A$ is simply connected. Am I correct?
If so, why is it never (or seldom) stated this way? The same applies to the residue theorem and other related theorems of complex analysis.

Related
I would be less surprised if the following statement was completely obvious:

Let $A\subset\mathbf{C}$ open, $I$ a compact interval, and $\gamma:I\rightarrow A$ a continuous path homotopic in $A$ to a point. Then there exists $U\subset A$ open and simply connected with $\gamma(I)\subset U$.

I suspect this is true, and I have a vague idea how to prove it. But it's not a one-liner and I don't remember seeing it in any textbook. Am I missing something?

REPLY [8 votes]: In fact, I would argue Cauchy's integral theorem in the proper form is the fact the following three statements are equivalent for $\gamma$ a loop in some open set $\Omega$:

The loop $\gamma$ is nullhomologous in $\Omega$, that is, it has index (or winding number) zero around every point in the complement of $\Omega$,
The induced map $f\in \mathcal O(\Omega)\longmapsto \int_\gamma fdz\in \mathbb C$ is identically zero,
For each $f\in \mathcal O(\Omega)$, Cauchy's formula holds for $\gamma$:
$$\frac{1}{2\pi i}\int_\gamma \frac{f(\xi)}{\xi-z}d\xi = I(\gamma,z) f(z).$$

Since all nullhomotopic loops are nullhomologous (by the homotopic invariance of the integral) you obtain your form of Cauchy's integral theorem. One argument in favour of putting Cauchy's theorem in the above form is that it can be restated in the following form: let $H_1(\Omega)$ be the space of chains in $\Omega$ (formal sums of loops) modulo nullhomologous chains, and let $H^1(\Omega)$ be the space of holomorphic functions modulo functions that are derivatives. Then there is a nondegenerate bilinear form
$$H_1(\Omega)\otimes H^1(\Omega)\to \mathbb C$$
that takes the class of a chain $\Gamma$ in $\Omega$ and the class of a holomorphic function $f$ in $\Omega$ and assigns it to $\int_\Gamma fdz \in \mathbb C$. This is a version of de Rham's theorem in this rather simple case.