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I just read about one point compactification and i am having some difficulty in grasping the concept.

Does one point compactification mean that we are simply adding a point to a non compact space to make it compact.

For example, my book says that $S^n$ is the one point compactification of $\mathbb R^n$, i don't quite follow this.

I need some help in understanding this concept, maybe with the help of the above stated example.

johny
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    Start with $S^2$. Can you visualize how to hover a point over the $\mathbb R^2$ plane and then, through the projective map, embed $\mathbb R^2$ into the sphere? – John Douma Sep 14 '15 at 15:10

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No, you don't just add a point. You have to define the topology in the newly created space (set). In case of $\mathbb{R}^n $ one thinks of this as adding a point at $\infty$. A topology may be defined by saying that set is, by definition, a neighbourhood of this point if it contains the exterior of some closed ball $B_R(0)$ (which can be thought of as balls around the point at infinity), or by defining a set to be open if it does not contain the point at infinity and is open in $\mathbb{R}^n $. In the general case the process is similar, in principle. See, e.g. Compactification

Sam Cappleman-Lynes
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Thomas
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