TITLE: To put a point at infinity
QUESTION [7 upvotes]: In physics, we say that we can put points at infinity in order to compactify some manifolds, like compactifying the plane via the stereographic projection. But, in order to do this, we also need to add a point at infinity.
To map the plane to the sphere we use the aforementioned projection. But, how do we naturally include the extra point at infinity? Do we do it in a forced way by just defining a point as the point at infinity? If so, how exactly?
Lastly, since infinity is not a number (so it cannot correspond to a point, but correct me if I am wrong), how can we assign a point to it? How can we intuitively think of adding a point at infinity?  
Thank you.

REPLY [6 votes]: Based on your comments, I think I can pinpoint some aspects which might interest you.
As was mentioned several times, the construction is simple: you pick a point $p$ not in the manifold, and define its neighbourhoods as complements of compact sets. The resulting space $\widetilde{M}=M \cup \{p\}$ with the topology generated by this process is called the one-points compactification of $M$ (or Alexandroff compactification).
This is a general process in a sense: one can prove that it works comfortably in any locally compact Hausdorff topological space which is not compact. However, there is an issue: it may result in something which is not a manifold, even if your original space was a manifold. 
Thus, your quest to search coordinates for this compactification in the comments may be in vain.
For example, consider two disjoint open intervals as the manifold $M$. The one-point compactification of such space is homeomorphic to a figure-eight, which is not a manifold. You may think that this is simply an artifact of $M$  not being  connected, but it isn't: take $M$ an open cylinder. The one-point compactification of it is homeomorphic to a horn torus, which is also not a manifold.