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$RP^n=S^n/{\sim}$ where $x\sim-x$. Let $p:S^n\rightarrow RP^n$ and this map is locally trivial with the fibre the two point set.

Say if $n=2$, then $RP^{2}$ will be lines passing through the origin in $R^3$ and $S^2$ is the sphere.

If this map is locally trivial, then it mean that for an open neighbourhood $U$ in the base space, the preimage $p^{-1}(U)$ behaves like $U\times F$ where $F$ is the fibre.

Visually what is this $U$ like? Is it like a "bundle" of lines through origin? Like two cones with their vertices touching each other? Also, how to visualize the fibre $F$ and why is the preimage (a region of the sphere I assume) like $U\times F$? What is this "two point set"?

Appreciate if anyone can help in the visualisation. Thanks!

Michael Hardy
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LanaDR
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  • Yes, you have good "visualization." It is a cone of lines through the origin (with the origin deleted). Draw a diameter through the sphere. Where this intersects the sphere, draw two tiny patches with antipodal symmetry. When you identify this in $\mathbb{R}P^2$, these will be a single open set. Thus, the preimage of that single open set is your two original tiny patches on the sphere, which is $U \amalg U \cong U \times {\pm 1}$. The fiber above $[x] \in \mathbb{R}P^2$ is the antipodal pair ${x, -x}$. – Randall Sep 12 '17 at 01:11
  • See my edits to the question for proper MathJax usage. $$ S^n/{\sim} $$ – Michael Hardy Sep 12 '17 at 02:41

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