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I have read the paper The Jordan-Brouwer Seperation Theorem written by Wolfgang Schmaltz. The main result in paper is below

Any compact, connected hypersurface $X$ in $\mathbb R^n$ will divide $\mathbb R^n$ into two connected regions; the outside $D_0$ and the inside $D_1$. Further more, $\bar{D_1}$ is itself a compact manifold with boundary $\partial\bar{D_1}=X$.

Now I am curious about the generalization of theorem. Can someone give any reference? Thank you in advance!

I also raise up one of my confusions:

Consider the noncompact case. That is to say, we only assume $X$ is complete and connected. So $X$ may be not compact.

Can the theorem above still be right, if $D_0$ and $D_1$ are both unbounded?

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    $\begingroup$ Do you know about Poincare-Alexander duality for cohomology with compact support? Using it you get the same JB-separation conclusion in the noncompact case (instead of "complete" one usually says "properly embedded"). $\endgroup$
    – Misha
    Apr 18, 2014 at 22:42
  • $\begingroup$ @Misha Is there any reference? Thank you. $\endgroup$
    – gaoxinge
    Apr 19, 2014 at 3:29
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    $\begingroup$ A. Hatcher's "Algebraic Topology". $\endgroup$
    – Misha
    Apr 19, 2014 at 16:40

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