I know that the real projective plane $P^2$ can be thought of as a union of a mobius band and a disk, where the union occurs among the common boundary of the two (circle).
My question is about $P^3$. I don't know how to describe it. Is it union of two manifolds as $P^2$, and if so what are they?
There is a very natural generalization of how you imagine $RP^2$:
Every projective space $RP^n$ has up to isomorphism a unique non trivial line bundle. This reduces to the Mobius band for $n=1$ and hence generalizes this notion. In fact we want to take the disk bundle $D(RP^n)$ here, so that we get boundary.
Now the boundary of $D(RP^n)$ is obviously connected, in fact, since the line bundle is non trivial, the boundary which is the corresponding sphere bundle doubly covers $RP^n$ by restricting the projection. Hence it is $S^n \cong \partial D^{n+1}$ and we obtain $RP^{n+1}$ as the union of $D(RP^n)$ and $D^{n+1}$ along the common boundary.