Here is what I got but my professor say it's wrong
Let $X$ be a compact, connected hyperspace in $R^n$, then $ R^n-X$ consist of 2 open sets $D_0$ – the outside component and $D_1$ the inside component. Moreover, $∂(\overline {D_1} )=X$. Let $z∈R^n-X$. Consider any ray from $z$ that is tranversal to $X$, if the ray intersect $X$ even number of times then $z$ is out side, otherwise, $z$ is inside. Since $z∈R^n-X$, and $z$ can lie outside or inside of $X$, this make $D_0$ and $D_1$ two connected component that contain every point $z∈R^n-X$.
If I'm not allowed to use the Jordan Brouwer theorem, then I guess the Alexander duality are out of the option as well. So how can I prove this problem without those?