0
$\begingroup$

How can I construct a continuous two-to-one map from the 2-sphere $S^2$ to the real projective plane?

I know that a real projective plane $\Bbb RP^2$ is described to be the quotient space of $S^2$ by identifying two points. But from this, how will I be able to construct a continuous map?

$\endgroup$
2
  • $\begingroup$ It's literally the quotient map $p:S^2\to\mathbb{R}P^2$. $\endgroup$
    – freakish
    Commented Jun 15, 2021 at 11:24
  • $\begingroup$ and what would be its elements? $\endgroup$
    – Roj_yel
    Commented Jun 15, 2021 at 11:26

1 Answer 1

2
$\begingroup$

The map takes the point $p \in S^2$ to the equivalence class $\{p, -p\}$ in $\Bbb RP^2$.

$\endgroup$
1
  • $\begingroup$ ohh. okay! got it. Thanks! $\endgroup$
    – Roj_yel
    Commented Jun 15, 2021 at 11:30

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .