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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?

XiaoXiao Zhen
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    When you define $D_0$ and $D_1$, you're begging the question. It looks like you mean to define $D_0 = {z\ |\ \mbox{every ray from $z$ intersects $X$ even number of times}}$ and $D_1$ for odd numbers of times. So now you have to show that the parity of the number of intersections doesn't depend on the ray you choose from $z$, that they're each open, that their union is dense, and that they're each connected. – Neal Nov 22 '14 at 17:06
  • Can you explain it a little bit clearer please? What do you mean by saying the parity of the number of intersection doesn't depend on on the ray from $z$. If it's start from $z$ in $D_0$ then the parity of intersection will different than when it start at $z$ in $D_1$, so it does depend on where the ray start, right? Or I misunderstand something? – XiaoXiao Zhen Nov 22 '14 at 17:25
  • There are many rays coming out of $z$. How do you know that all of the transverse intersections with $X$ are even (or odd, as the case may be)? – Neal Nov 22 '14 at 18:52
  • because the rays is kinda homotopic to each other, if they are homotopic and tranverse intersect with $X$ then they must have the same number of intersection, and the number of the intersection can only be either even or odd? Is that what you are saying? – XiaoXiao Zhen Nov 22 '14 at 20:33
  • No. Here are three questions to answer precisely. Why can $z$ not have one ray that intersects $X$ transversely an odd number of times and another ray that intersects $X$ transversely an even number of times? Why must $D_0$ and $D_1$ be open with dense union? Why are $D_0$ and $D_1$ each connected? – Neal Nov 22 '14 at 20:53

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