I am working over the paper: [Target Enumeration via Euler Characteristic Integrals](http://epubs.siam.org/doi/abs/10.1137/070687293) and in order to follow a proof I need to prove:



> 
> If $A$ is compact nonempty subset of $\mathbb{R}^2$, then the singular homology
>  groups of $A$, $H\_k(A)$ vanish for $k\geq 2$.
> 
> 
> 


The result seems seems reasonable for me (there are non obvious counterexamples in dimension grater than three but not in dimension two). But I am having difficulties finding a proof. 


What I have tried or thought so far:


* Using the long exact sequence in homology and it didn't work.
* Maybe trying luck with de Rham cohomology and using some kind of isomorphism followed by the Universal Coefficients theorem in cohomology....
* Using some dimension theory but I have no clue how...


Any ideas? Thanks in advance and any help would be appreciated.



Homology groups of compact subset of $\mathbb{R}^2$
This is answered in the paper



> 
> *The singular homology group of planar sets do not behave anomalously* 
>  by Andreas Zastrow
> 
> 
> 


This appears to be a link to the paper: <http://at.yorku.ca/i/d/e/b/11.htm>


