Q:

What are the compactifications of those spirals?

I want to find the explicit compactification (I don't mind if its a one-point compactification or not) of the spiral 
$E_{1}=\left\{ \left(e^{t}\cos{t},e^{t}\sin{t}\right):t\in\mathbb{R}\right\} .$
and the spiral 
$E_{2}=\left\{ \left(\frac{t}{t+1}\cos{t},\frac{t}{t+1}\sin{t}\right):t\in\left(0,+\infty\right)\right\}$ 

What I've managed to do is... for $E_{1}$:
First of all, I thought that I had to add $\left(0,0\right)$ because the limit when $t\rightarrow$-$\infty$ doesn't reach it, so the $\infty$ has to be linked to the origin, and because of the fact that $f\left(t\right)=\left(e^{t}\cos{t},e^{t}\sin{t}\right)$ is a parametrization, so it is an homeomorphism to $\mathbb{R}^{2}$ so the compactification that comes to me is the one that comes with the map $f\left(t\right)=\begin{cases}
\left(e^{t}\cos{t},e^{t}\sin{t}\right) & t\in\mathbb{R}\\
\left(0,0\right) & t=-\infty
\end{cases}$ but $\partial E_{1}$ is what fails. So I don't think that I could the next one.
Thanks in advance!

A:

A compactification of $E_2$ is $E_2 \cup S^1.$

