There's a theorem in mathematics that states:
Let $A$ be an open path connected set in $\mathbb{R^m}$, and $\omega:A\to\mathbb{R^{1\times2}},x\to[\omega^1(x)\space...\space \omega^m]$ be a 1 degree differential form, then $\omega$ is exact.
But we know that: $$\omega:\mathbb{R^2\setminus \{(0,0)}\}\to \mathbb{R^{1\times2}}\\(x,y)\to\bigg[ -\dfrac{y}{x^2 + y^2} \space\ \dfrac{x}{x^2 +y^2}\bigg]$$
is not an exact differential form, however, $\mathbb{R^2}\setminus \{(0,0)\}$ is an path connected set. What am I getting wrong?
PS:I may be a bit lost in translation about some technical terms, and some rigor may also be flawed, but I hope the message gets through.