I am trying to calculate $H_k(X)$ where $X = \mathbb{R}P^n - \{ x_0 \}$
I started thinking about $k=2$. We can get the projective plane by taking the upper hemisphere with points along the equator identified according to the antipodal map. If we remove a point from the hemisphere, we can then enlarge this such that we are just left with a circle and thus for $k=2$ we just have the homology of the circle.
My geometric intuition starts to fail for $k=3$ and higher spaces. So my questions are:
1) Does the same construction work for higher $k$? (probably not, this seems too easy)
2) If not, what is the nice way to calculate the homology groups (say we know $H_k(\mathbb{R}P^n)$? I guess there is a way to use Mayer-Vietoris, but I just can't see it