Geodesic
$R^{(\infty)}$ is diffeomorphic to $R^{(\infty)}\setminus\{0\}$
Matematičeskie zametki, Tome 20 (1976) no. 5, pp. 741-746.

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We construct a one-to-one map of the topological direct sum $R^{(\infty)}$ of countably many copies of the real line onto $R^{(\infty)}\setminus\{0\}$ which is infinitely differentiable along with its inverse, in the sense of Michael–Bastiani. 
@article{MZM_1976_20_5_a13,
     author = {A. A. Lobuzov},
     title = {$R^{(\infty)}$ is diffeomorphic to $R^{(\infty)}\setminus\{0\}$},
     journal = {Matemati\v{c}eskie zametki},
     pages = {741--746},
     publisher = {mathdoc},
     volume = {20},
     number = {5},
     year = {1976},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1976_20_5_a13/}
}
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A. A. Lobuzov. $R^{(\infty)}$ is diffeomorphic to $R^{(\infty)}\setminus\{0\}$. Matematičeskie zametki, Tome 20 (1976) no. 5, pp. 741-746. http://geodesic.mathdoc.fr/item/MZM_1976_20_5_a13/