Questions.

(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:

(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.

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:

(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.

Questions.

(0) How would you prove, using classical logic and contemporary 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:

(Own work; using SageMath, the Sage Mathematics Software System, retouched with GIMP.)

• 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.

Questions.

(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:

(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.