I have some trouble with a problem set. It is specifically part (i) that I am stuck on (I have left in part (ii) for context):
Let $M$ be an $n$-dimensional manifold and let $A$ be an abelian group. For a point $x \in M$, we write $\rho_x : H_n(M; A) → H_n(M|x; A) = H_n(M, M \setminus \{x\}; A)$ for the canonical map from the long exact sequence of the pair $(M, M \setminus \{x\})$.
(i) Let $\mu \in H_n(M; A)$ be a homology class. Show that the subset $$\{x \in M | \rho_x (\mu) =0\}$$ is both open and closed.
(ii) Show that $H_n(M; A)$ is trivial if M is connected and non-compact.
The fact that this subset is both open and closed makes it obvious it must either be empty or $M$. Clearly if $\mu =0$ we have that this set is the whole of $M$. I just have no idea on how to proceed.
This exercise corresponds to a section on $R$-orientations (Poincaré duality was not covered up to this point).