As an excercise, I have to give 6 compact, Hausdorff and path connected topological spaces ${Z_k}_{k=1}^6$ such that pairwise are not homotopic and satisfying that $|\pi_1 (Z_k) | \leq 5$.
My question is, is it fine to take $Z_k = {\Bbb{P} \Bbb{R}}^{k+1}$ ? They are compact, Hausdorff and path connected, but I do not know how to justify that they are pairwise not homotopic