Denote by $C^r_S(\mathbb R,\mathbb R), C^r_W(\mathbb R,\mathbb R)$ for the $C^r$-strong and $C^r$-weak toplogy respectively(c.f. http://planetmath.org/mathcalcrtopologies). An exercise in M. Hirsch's Differential Topology asks to prove $C^{r+1}_S (\mathbb R,\mathbb R)\approx C^r_S(\mathbb R, \mathbb R)\times\mathbb R$ and $C^{r+1}_W (\mathbb R,\mathbb R)\approx C^r_W(\mathbb R, \mathbb R)\times\mathbb R$. I suppose this is an easy exercise, since it's not starred in the book. However, I found the statement hard to imagine. Since there is a natural embedding $C^{r+1}_S(\mathbb R,\mathbb R)\hookrightarrow C^r_S(\mathbb R,\mathbb R)$, the statement would imply that $C^r_S(\mathbb R,\mathbb R)\times \mathbb R$ can be embedded into $C^r_S(\mathbb R,\mathbb R)$, which is, though acceptable, quite counterintuitive for me. For this reason, I can't even find a reasonable bijection(except for the one induced by the same cardinality) between the two spaces, let alone a homeomorphism.
So my question is:
Is there a way to think of the statement intuitively? How to prove this statement?
Any help is appreciated.