I have a question related to the notion of extended real line. I am a very beginner of this topic and in what follows I might say things that look no-sense for an expert in the field.
The extended real line is the real line $\mathbb{R}$ plus $+\infty$ and $-\infty$. Is there a similar notion for an Euclidean space $\mathbb{R}^k$ with $k>1$?
There are two "natural" infinities in the real line, due to it being totally ordered. In higher dimensions, we usually add one point at infinity, in a process that can be applied to any space (it is called the one-point compactification.)
Topologically, the one-point compactification of $\mathbb R^n$ can be see as an $n$-sphere.
For example, the one-point compactification of the real line is a circle (because we've essentially joined $-\infty$ and $+\infty$ into one point.)
There is another way to add infinities to $\mathbb R^n$: $\mathbb RP^n$ is the $n$-dimensional projective space, and it has a point at infinity for every class of parallel lines in $\mathbb R^n$.
As a topological space, $\mathbb RP^n$ is a bit odder, but it can be seen as an $n$-sphere with anti-podal points identified. When $n=1$, this is still just a circle, but when $n>1$, you get a space that is not "orientable," much like the Möbius strip.