I'm working on a proof to show there exists an embedding of the real projective plane P R2 in R4.(adsbygoogle = window.adsbygoogle || []).push({});

The initial setup is as follows:

Let S2 denote the unit sphere in R3 given by S2 = {(x, y, z) ∈ R3 : x2 + y 2 + z 2 = 1}, and let

f : S2 → R4 be deﬁned by f (x, y, z) = (x2 − y 2 , xy, yz, zx).

I'm trying to show that f determines a continuous map F: P R2 → R4 where P R2 is the real projective plane,

then show that F is a homeomorphism onto a topological subspace of R4 .

I think it's easy to see that f(x1,y1,z1)=f(x2,y2,z2) implies (x1,y1,z1)=+/-(x2,y2,z2). But I don't know how to figure out the whole proof completely. Could anyone please give me a hint? Any input is appreciated!

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# Real projective plane

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