This map is an isomorphism and it is called the Alexander-Čech duality (or sometimes simply Alexander duality). It can be considered for infinite complexes as well, but in that case one has to change the flavour of (co)homology theories involved. $H_i$ is then the Steenrod-Sitnikov homology and $H^{n-i}$ has to be cohomology (?).

The Spanier-Whitehead duality is a generalization, where $S^{n+1}\backslash X$ is replaced by any space $D_n(X)$, together with an element $\delta$ such that the corresponding map

$\delta_/ : H_i(X) \to H^{n-i}(\mathcal{D}_n X)$

is an isomorphism. It follows that one may replace $\mathcal{D}_n X$ by its suspension and so on, hence the stable homotopy theory is a natural setup for this duality.