For the sake of completeness, let's look at the following simple example of very *nice* continua, and of their inverse system of inclusions, for which there is no **homological** surjection.

All continua are subspaces of $\ \mathbb R^2$. Let $\ X:=(0\ 0)\ $ be a single-point space. Let $\ C(p;r)\ $ and $\ B(p;r)\ $ stand for closed and open discs which have their center in $\ p\ $ and their radius equal to $\ r.\ $ Define:

$$X_n\ \,:=\ \,C\left(\left(2^{-n}\ 0\right);\ 2^{-n}\right)\ \setminus\ B\left(\left(\frac 3{2^{n+1}}\ 0\right);\  2^{n+1}\right)$$

Thus $\ H_*(X)=0\ $ while $\ H_*(X_n) = H_*(S^1).\ $ As we see, there are no surjections.(homological)