I computed some homology groups, could you tell me if I got it right? Thanks.
(i) Let $M$ be the Möbius band. Then $H_0(M, \mathbf{Z}_2) = H_1(M, \mathbf{Z}_2) = \mathbf{Z}_2$ and $H_n(M, \mathbf{Z}_2) = 0$ otherwise.
(ii) $H_0 ( S^1, \mathbf{Z}_2) = H_1(S^1, \mathbf{Z}_2) = \mathbf{Z}_2$ and $H_n(S^1, \mathbf{Z}_2) = 0$ otherwise.
So we cannot distinguish between the Möbius band and $S^1$ in $\mathbf{Z}_2$, even though they are not homeomorphic.
(iii) $H_0 (S^2, \mathbf{Z}_2) = \mathbf{Z}_2 $, $H_2 ( S^2, \mathbf{Z}_2) = \mathbf{Z}_2$ and $H_n(S^2, \mathbf{Z}_2) = 0$ otherwise.
(iv) Let $T = S^1 \times S^1$. Then $H_0(T, \mathbf{Z}_2) = \mathbf{Z}_2$, $H_1(T, \mathbf{Z}_2) = \mathbf{Z}_2 \oplus \mathbf{Z}_2$ and $H_2(T, \mathbf{Z}_2) = \mathbf{Z}_2$. $H_n(T,\mathbf{Z}_2) = 0$ otherwise.
So we see that we can still distinguish some non-homeomorphic spaces.