Ordinary cohomology on CW complexes is determined by the coefficients. There are (more than) two nice ways to define cohomology for non-CW-complexes: either by singular cohomology or 
by defining $\widetilde H^n(X;G) = [X, K(G,n)]$. Are there standard/easy examples where these
two theories differ?


One idea that comes to mind is the paper by Milnor and Barratt (about Anomolous Singular
Homology) which says that the $n$-dimensional Hawaiian earring $H^n$ has nontrivial singular
homology in arbitrarily high dimensions. But I don't see an easy way to compute
$[H^n, K(G, m)]$.



Difference between represented and singular cohomology?
The Cantor set has exotic zeroth cohomology. Its singular cohomology is the linear dual of its zeroth singular homology, which is the free abelian group on its set of points. Thus its singular cohomology is an uncountable infinite product of $\mathbb Z$. Its represented cohomology is the set of continuous maps to the discrete space $\mathbb Z$, which must factor through a finite quotient. It is a free abelian group on countably many generators.


