**Questions.**

EDIT: readers please note that while this question arose in research, the OP was so hung-up on a question concerning infinite planar graphs that a *strong a-forteriori-reason*, kindly pointed out by the answerer below, for why (A) is true was overlooked. The question as it stands is *trivial*. It will perhaps be *edited to include the (still open) graph theoretical question*, in due course. END OF EDIT

(0) How would you prove, in  *usual topology*, the following assertion: 

(A) There does not exist any plane isotopy carrying the subset $S=\{ (\exp(t)\ \cos(t),\exp(t)\ \sin(t)|\quad t\in\mathbb{R}\}\subseteq\mathbb{R}^2$ 
onto the subset 
$R=\{ (t,0)|\quad t\in\mathbb{R}\}\subseteq\mathbb{R}^2$.

(1) Independent of (0), in a bibliographic/reference-requestish vein: 

(B) In what *literature references* does (something equivalent to) (A) *recognizably* appear? (I'm interested in as many relevant references as possible, in any of the media: book, lecture notes, research paper, website.)

**Remarks.**

 * In $A$, all technical terms are standard terms of basic topology nowadays. The term *plane isotopy* would in some contexts often called by the more general term *ambient isotopy*. For what it's worth, a definition of the central notion here is the following.

Let 

$\eta_S\colon \mathbb{R}\rightarrow\mathbb{R}^2$ be given by $\eta_S(t) =  (\exp(t)\cdot\cos(t),\exp(t)\cdot\sin(t))$,

and 

$\eta_R\colon \mathbb{R}\rightarrow\mathbb{R}^2$ be given by $\eta_R(t) = (t,0)$.

Then a *plane isotopy* is any continuous set-map 

$\theta\colon \mathbb{R}^2\times[0,1]\rightarrow\mathbb{R}^2$

satisfying the three axioms

(A.0) for all $v\in\mathbb{R}^2$, $v=\theta(v,0)$,

(A.1) for all $t\in[0,1]$, $v\mapsto\theta(v,t)$ defines a homeomorphism $\mathbb{R}^2\rightarrow\mathbb{R}^2$,

(A.2) $(v\mapsto \theta(v,1))\circ\eta_S = \eta_R$, equal as set-maps.


 * I am less, but also, interested in *the* correct answer here, more in different writing- and proof-styles, some more efficient than others, localized at this very question. 


 * Of course, $S\subseteq\mathbb{R}^2$ 'looks' something like the blue line in the following illustration:
[![enter image description here][1]][1]

(Made with Sage.)

 * Motivation for this question is that (A) came up in research about (three-connected) *infinite planar graphs*, and I need to know more about and around it.


  [1]: https://i.sstatic.net/jC9RN.jpg