To put a point at infinity 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.
 A: One way to do it consists in considering the (affine) real plane as the (affine) complex line.
A point on this line is just a given complex number $z$. The one-point compactification is obtained considering the (affine) complex plane $\mathbf C^2$. We have a map
\begin{align}\mathbf A^1(\mathbf C)=\mathbf C&\longrightarrow \mathbf A^2(\mathbf C)=\mathbf C^2\\
z&\longmapsto (z,1)\end{align}
Now we define an equivalence relation on $\mathbf C^2\smallsetminus\bigl\{(0,0)\bigr\}$:
$$(z,u)\sim (z',u')\overset{\text{def}}{\iff}\exists\lambda\in\mathbf C^*, z'=\lambda z, u'=\lambda u.$$
This means all points in $\mathbf C^2$, different from the origin, in the complex subspace $\langle(z,u)\rangle$ are equivalent to $(z,u)$.
The projective complex line $\;\mathbf P^1(\mathbf C)$ is, by definition, the quotient set $ \bigl(\mathbf C^2\smallsetminus\bigl\{(0,0)\bigr\}\bigr)/\sim$, endowed with the quotient topology. One shows $
\mathbf P^1(\mathbf C)$ is a compact space.
A point in $\;\mathbf P^1(\mathbf C)$ is denoted $[z:u]$. The above map, composed with the canonical map 
\begin{align}
\mathbf C^2\smallsetminus\bigl\{(0,0)\bigr\}&\longrightarrow\mathbf P^1(\mathbf C)\\
(z,u)&\longmapsto [z:u]
\end{align}
 yields the map
\begin{align}
\mathbf A^1(\mathbf C)\bigr\}&\longrightarrow\mathbf P^1(\mathbf C)\\
z&\longmapsto [z:1],
\end{align}
and any (projective) point$[z:u]$ such that $u\ne 0$ is the canonical image of the unique point $z/u\in \mathbf A^1(\mathbf C)$, since $\;(z,u)\sim (z/u,1)$.
What if $u=0$? All points $(z,0)$ are equivalent, hence yield a unique  point in $ \mathbf P^1(\mathbf C)$. We call it the point at infinity of the affine (complex) line.
To sum up:
$$\mathbf P^1(\mathbf C)=\mathbf A^1(\mathbf C)\cup\{\infty\}.$$
A: How do you throw a bunch of paper together into some binding and get a book? How do you toss a fish into a bowl of water and get a fish bowl? How do you add espresso into some hot milk and get a latte?
The thing you throw in has nothing to do with the original collection. At all. And you're not trying to interpret the thing you added as somehow "the same" as the collection that was originally there. But the thing you throw into the collection creates a new thing altogether that gets its own new name.
The "point at infinity" is the binding, the fish, the espresso. The "one-point compactification" is the book, the fish bowl, the latte.
A: 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.
A: Topologically, the way to put a point somewhere in particular is to define which sets are its neighborhoods. If you know all the open sets containing a point, then you know "where" the point is.
To add a point at infinity, the usual trick is to say that we adjoin one point to the underlying set, and then define its open neighborhoods as the complements of existing compact sets. This trick captures the intuitions that we want to have about "infinity". Think of it this way: exteriors of larger and larger discs centered at the origin (or anywhere else) are smaller and smaller neighborhoods of infinity. This captures the idea that compact sets are bounded away from infinity.
If you project the plane onto a sphere via the stereographic, then you can see that exteriors of such circles actually map to smaller and smaller open discs around the missing point.
