Projective Plane and Projective Space

I have already heard of a $n$ dimensional manifold called the projective space which is the set of all lines through the origin of $\mathbb{R}^{n+1}$. Spivak presents in his Differential Geometry book another manifold called the projective plane which is obtained by identifying antipodal points on the sphere with center at the origin.

My question is: are those spaces equal up to homeomorphism (obviously setting the correct dimensions) or it's just that their names are almost the same? I have no idea on how to tackle this, however I thought that they really should be homeomorphic, because if I take $p \in S^2$ and identify with $-p\in S^2$ in practice I'm getting points that lines on some line through the origin, so I imagine I can make a map $f$ from the projective plane to the projective space by sending each $\{p, -p\}$ from the projective plane to the line through the origin that contains it in the projective space.

I don't know if this map is homeomorphism, however I can see intuitively that it's bijective. Can someone give some help with this? Are those spaces homeomorphic, or it's just that their names are almost equal but referencing totally unrelated things?

Thanks very much for the help.

Indeed, the projective plane is the space of all lines (through the origin) in $\mathbb{R}^3$, each of which intersects the two-sphere in antipodal points. Thus there is a one to one correspondence between lines through the origin in $\mathbb{R}^3$ and antipodal points of the two-sphere.