Real Line: Topology Problem
Given the real and positive line $V=\mathbb{R},\mathbb{R}_+$,
Regard product structures:
$$\pi^\lambda:V^\Lambda\to V:(x_\lambda)_{\lambda\in\Lambda}\mapsto x_\lambda$$
And characteristic properties:
$$\chi:\Omega\to V^\Lambda:\quad\chi^{-1}\left(\mathcal{T}V^\Lambda\right)\subseteq\mathcal{T}\Omega\iff(\pi^\lambda\circ\chi)^{-1}(\mathcal{T}V)\subseteq\mathcal{T}\Omega\quad(\lambda\in\Lambda)$$
Regard algebraic structures:
$$\alpha:V\times V\to V:(x,y)\mapsto x+y$$
$$\mu:V\times V\to V:(x,y)\mapsto xy$$
And impose continuity:
$$\alpha^{-1}(\mathcal{T}V)\subseteq\mathcal{T}(V\times V)$$
$$\mu^{-1}(\mathcal{T}V)\subseteq\mathcal{T}(V\times V)$$
Regard injection and projection:
$$\iota:\mathbb{R}_+\hookrightarrow\mathbb{R}:x\mapsto x$$
$$\pi:\mathbb{R}\twoheadrightarrow\mathbb{R}_+:x\mapsto|x|$$
And characteristic properties:
$$\eta:\Omega\to\mathbb{R}_+:\quad\eta^{-1}(\mathcal{T}\mathbb{R}_+)\subseteq\mathcal{T}\Omega\iff(\iota\circ\eta)^{-1}(\mathcal{T}\mathbb{R})\subseteq\mathcal{T}\Omega$$
$$\vartheta:\mathbb{R}_+\to\Omega:\quad\vartheta^{-1}(\mathcal{T}\Omega)\subseteq\mathcal{T}\mathbb{R}_+\iff(\vartheta\circ\pi)^{-1}(\mathcal{T}\Omega)\subseteq\mathcal{T}\mathbb{R}$$
Moreover impose Hausdorff!

Does this give euclidean topology?

Application
This is a lemma for: Dual Space
 A: Meanwhile I got it.. :D
Product Topology
For the product topologies:
$$\mathcal{T}V^\Lambda=\left\langle\bigcup_{\lambda\in\Lambda}(\pi^\lambda)^{-1}(\mathcal{T}V)\right\rangle$$
(This result is independent!)
Real & Positive Line
As it is Hausdorff TVS:*
$$\dim\mathbb{R}<\infty:\quad\mathcal{T}\mathbb{R}=\mathcal{T}\mathbb{R}_E$$
That gives rise to:
$$\mathcal{T}\mathbb{R}_+=\iota^{-1}(\mathcal{T}\mathbb{R}_E)$$
Hausdorff transfers as:
$$\iota(a)=\iota(a')\implies a=a'$$
(This proves uniqueness!)
Injection & Projection
As they are inverses:
$$\pi\circ\iota=\mathrm{id}_+:\quad\pi^{-1}(N)\in\mathcal{T}\mathbb{R}_E\implies N\in\iota^{-1}(\mathcal{T}\mathbb{R}_E)$$
Conversely they act as:
$$\iota^{-1}(A)=A\cap\mathbb{R}_+\quad\pi^{-1}(B)=B\cup(-B)$$
For nontrivials one has:
$$\mathbb{R}_+\in\mathcal{N}_a\mathbb{R}_E\quad(a\neq0)$$
So on neighborhood filters:
$$N\in\iota^{-1}(\mathcal{N}_a\mathbb{R}_E)\implies N=M\cap\mathbb{R}_+\in\mathcal{N}_a\mathbb{R}_E\\
\implies N\cup(-N)\in\mathcal{N}_a\mathbb{R}_E\implies\pi^{-1}(N)\in\mathcal{N}_a\mathbb{R}_E$$
For the trivial one has:
$$\mathcal{B}_0\mathbb{R}_E:=\{(-\varepsilon,\varepsilon):\varepsilon>0\}:\quad\langle\mathcal{B}_0\mathbb{R}_E\rangle=\mathcal{N}_0\mathbb{R}_E$$
So on neighborhood filter:
$$N\in\iota^{-1}(\mathcal{N}_0\mathbb{R}_E)\implies N=M\cap\mathbb{R}_+\supseteq[0,\varepsilon)\\
\implies N\cup(-N)\supseteq(-\varepsilon,\varepsilon)\implies\pi^{-1}(N)\in\mathcal{N}_0\mathbb{R}_E$$
(So they are compatible!)
Addition & Multiplication
Introduce the mapping:
$$\Phi:\mathbb{R}_+\times\mathbb{R}_+\to\mathbb{R}\times\mathbb{R}:(x,y)\mapsto(x,y)$$
It is continuous since:
$$\pi^\lambda\circ\Phi=\iota\circ\pi_+^\lambda\in\mathcal{C}(\mathbb{R}_+\times\mathbb{R}_+,\mathbb{R})$$
So the operations as well:
$$\iota\circ\alpha_+=\alpha\circ\Phi\in\mathcal{C}(\mathbb{R}_+\times\mathbb{R}_+,\mathbb{R})$$
$$\iota\circ\mu_+=\mu\circ\Phi\in\mathcal{C}(\mathbb{R}_+\times\mathbb{R}_+,\mathbb{R})$$
(This proves existence!)
*Here's flaw: See comments!
