I am working over the paper: Target Enumeration via Euler Characteristic Integrals 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 full of common sense for me. 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.