I'd like to use the formulation of Lefschetz duality stated here, but I can't seem to find a reference for this particular version of it, and it doesn't seem quite right to me.
The exact statement in question is: Let $X$ be a Hausdorff topological space, and let $A \subset X$ be a subspace such that the complement $X - A$ is an orientable topological $n$-manifold. Then, for any abelian group $G$ and any $i$, $$H_i(X, A; G) \cong H_c^{n - i}(X - A; G),$$ where $H_i$ and $H_c^{n - i}$ denote singular homology and compactly supported singular cohomology, respectively.
The problem is, by Poincaré duality, $H_c^{n - i}(X - A; G) \cong H_i(X - A; G)$, so if the above statement is true, then $H_i(X - A; G) \cong H_i(X, A; G)$. But that definitely doesn't seem right; for example, if $X = \mathbb{R}^2$ and $A = \{(0, 0)\}$, then $H_1(X - A; G) \cong G$ because $X - A$ is homotopic to a circle, but $H_1(X, A; G) \cong \tilde{H}_1(X; G) = 0$.
Is the above statement true in this generality? If so, where can I find a proof, and where is my attempt at a counterexample mistaken? If not, what's the correct formulation?
(The main case I'm interested in is where $X$ is a complex projective variety and $A$ is a closed subvariety containing the singular locus of $X$.)