Let $f \colon X \to Y$ be a weak homotopy equivalence between topological spaces. If I am not mistaken, then one can rephrase this by stating that the induced map $[K,X] \to [K,Y]$ between homotopy classes of maps is a bijection for every finite simplicial complex $K$, right?

Now in a paper that I was reading the authors proved the same bijection, but for any compact Hausdorff space $K$, and I was wondering: How much more general than a weak homotopy equivalence is this?

(Note that in the paper the spaces $X$ and $Y$ are point-set topologically not well-behaved; especially, they are not CW-complexes. So one can not just conclude that they are homotopy equivalent to each other.)

One can phrase my question a bit more concretely in the following way: If the induced map $[K,X] \to [K,Y]$ is a bijection for every finite simplicial complex, and $X$ and $Y$ are CW-complexes (or more generally, have the homotopy type of CW-complexes), then $X$ and $Y$ are actually homotopy equivalent to each other. So if we can now prove that $[K,X] \to [K,Y]$ is a bijection for every compact Hausdorff space $K$, what kind of topological spaces can $X$ and $Y$ be in order to conclude that they are already homotopy equivalent?