The Borsuk-Ulam theorem states that if $f$ is a continuous function from the $n$-dimensional sphere $S^n$ to the $n$-dimensional euclidean space $\mathbb{R}^n$, then there are two antipodal points of $S^n$ which map to the same point.
I would like to apply this to a function $f$ which maps some points of $S^n$ to points at infinity.
Is the theorem still true if we replace $\mathbb{R}^n$ with the real projective space $RP^n$?
I tried searching for "Borsuk-Ulam real projective space" online but only came across things which go way over my head. As I am not a mathematician (nor math student) I'd prefer answers which stick to the basics and do not go to deep into stuff which I'm likely not to understand with my current level of mathematical sophistication (cohomology, fiber bundles, CW-complex, etc.) I would also be happy with references to introductory material which could answer my question (or help me answer it for myself).
Best Answer
This is not true. A projective line is homeomorphic to a circle.
Map points $f: S^1 \rightarrow RP^1: (\cos(\theta), \sin(\theta) ) \mapsto [\cos(\frac{\theta}{2}): \sin(\frac{\theta}{2})]$ This is a bijection since angles that differ by $\pi$ represent the same point in $RP^1$. It is also a homeomorphism with a continuous inverse.
Clearly this disproves the idea that there would be antipodal points mapped to the same point for any continuous map.