Let $\mathbb{R}P^m$ be the $m$-dimensional real projective space and let $\mathbb{R}P^m\setminus\{\*\}$ be the punctured space. I observe:


1. $\mathbb{R}P^2\setminus\{\*\}$ is homeomorphic to a (open) Mobius strip, that is, the total space of a non-trivial line bundle over $\mathbb{R}P^1$.
2. $\mathbb{R}P^1\setminus\{\*\}$ is homeomorphic to $\mathbb{R}$, which can be regarded as $\mathbb{R}P^0\times\mathbb{R}$.


Could I generalize (1) and (2) to the following?


(i). If $m$ is odd, then $\mathbb{R}P^{m+1}\setminus\{\*\}$ is homeomorphic to the total space of a non-trivial line bundle over $\mathbb{R}P^{m}$. 


(ii). If $m$ is even, then $\mathbb{R}P^{m+1}\setminus\{\*\}$ is homeomorphic to $\mathbb{R}P^{m}\times\mathbb{R}$. 


**Question:** Whether are (i) and (ii) true or false? Are there any related references?



homeomorphism type of punctured real projective spaces
(i) is true and (ii) is false. In fact, (i) is true for all $m\ge1$. More precisely, dropping a coordinate yields a map from $\mathbb RP^{m+1}\setminus\{\*\}\to\mathbb RP^m$ which is a locally trivial bundle with fiber $\mathbb R$. This bundle is not trivial for $m\ge1$ because, e.g., $\mathbb R P^m$ is orientable if and only if $m=0$ or $m$ is odd.


