I just read about the one-point, or Alexandroff, compactification of a topological space. It seems to be nice in its simplicity and easy description, but does suffer from some unfortunate properties like not being Hausdorff unless the original space is Hausdorff and locally compact.
My questions are: why study the one-point compactification? What is it useful for? Are there situations where we can learn more about the original space by instead passing to its one-point compactification and studying that?
I see the interest in the one-point compactification, and think it's a nice, intuitive idea. Normally I don't need a lot of motivation for various ideas and constructions and can appreciate them for what they are. In this case, however, I happened to find myself questioning what its real value in the 'big picture' is.