Is $\mathbb{R}^{\infty}$ homeomorphic to $\mathbb{R}^{\infty}\setminus\{0\}$? Let $\mathbb{R}^{\infty}$ be a linear topological space of all sequences
$x=(x_{1},x_{2},\ldots,x_{n},\ldots)$ of real numbers with a product topology,
or, in other words, let $\mathbb{R}^{\infty}$ be a countable product of real lines.
Is $\mathbb{R}^{\infty}$ homeomorphic to $\mathbb{R}^{\infty}\setminus\{0\}$,
where $0=(0,0,\ldots,0,\ldots)$?
If the answer is yes, how to prove this theorem?
Thank you in advance!

The answer is yes.
In Chapter VI § 2 of Selected Topics in Infinite Dimensional Topology (Bessaga & Pełczyński) is stated the surprising result that  $\mathbb{R}^\infty$, the Hilbert space $\ell_2$ and the unit sphere $S \subset\ell_2$ are homeomorphic spaces (${}^*\!$). The background required (developed in the previous chapters) seems to be rather wide. Anyway, proving this result, they exhibit an explicit homeomorphism between $S \smallsetminus \{\ast\}$ and $\ell_2$.
(${^*}\!$) moreover there is a 1966 paper by Bessaga entitled  Every infinite-dimensional Hilbert space is diffeomorphic with its unit sphere
