In the theorem and proof, we have seen that it requires the space to be finite-typed. To me, it is somehow "embarrassing" to have such a restriction...
In the proof, the key step is $$\text{Hom}(E_\cdot,\mathbb{Z})\otimes A\cong\text{Hom}(E_\cdot,A)$$ Where we require $E_{\cdot}$ to be finite-typed. Is there any way to remove the restriction and get the theorem for all topological spaces? E.g. accept some stronger axioms?