Is the extended $\overline{\mathbb R}^2$ written as $\mathbb R^2 \cup \{-\infty,+\infty\}$ or $(\mathbb R\cup \{-\infty,+\infty\})^2$ ? What is the name for the latter set?
A.k.a Are there four infinite points or just two or infinite number of infinite points?
Thanks for the information. So the most common compactification used is the Riemann sphere $\overline{\mathbb C}$ which adds only one point to $\mathbb C$.
The question I am more interested in is the natural generalization of the two points compactification of $\mathbb R$ into multiple dimensions. Intuitively, it should add more than four points to $\mathbb R^2$. It is not the Real projective space $\mathbb R\mathbb P^n$. Is it related to any named compactification? It has a poor algebraic structure, i.e. not a group or semi-group, so it might attract less interests.
Related questions:
Is there a name for complex numbers over affinely extended reals?
Do the two types of extended real number systems originate in algebraic geometry?
There are many compactifications of $\mathbb R ^n$. Perhaps the most common is the one-point compactification $\mathbb R ^n \cup \{\infty\}$ which is homeomorphic to the sphere $S^n$. You may also take $\overline{\mathbb R}^n$ whose remainder is homeomorphic to $S^{n-1}$. Another variant is $\overline{\mathbb R}^{n-m} \times (\mathbb R \cup \{\infty\})^m$.
If you are interested in compactifications $C$ of $\mathbb R ^n$ such that the remainder $C \setminus \mathbb R ^n$ is finite, then the answer is simple (although the proof is non-trivial): For $n \ge 2$ the one-point compactification is the only such compactification. See my answer to Is the two-point compactification the second-smallest compactification?.