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:

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

$\mathbb{R}P^1\setminus\{*\}$ 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?