**Theorem** $S^{n-1}$ disconnects $S^n$ into two open connected components, which have $S^{n-1}$ as frontier.



> 
> In $R^3$, if we replace sphere of standard torus with genus $g\geq1$, we may have "The Jordan-Brouwer Separation Theorem" intuitively. Then what happens when we replace topological sphere of topology torus with genus $g\geq1$?
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> We have $T^n$ in $R^{n+1}$, so do we have the same theorem?
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> What is the situation ahout manifold?
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**Theorem** $H\_k(S^n-S^r)=R,k=n-r-1$.



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> Does this theorem have the corresponding generalization?
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> 



The Jordan-Brouwer Separation Theorem
The magic words are: [Alexander duality.](http://en.wikipedia.org/wiki/Alexander_duality)


