Motivation for the one-point compactification
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.
Solution 1:
What we actually care about is relationships between topological spaces, and "$A$ is the one-point compactification of $B$" happens to be a particularly nice relationship about which it is possible to say a lot. For example, the sphere $S^n$ is the one-point compactification of $\mathbb{R}^n$, and this observation makes it possible to prove things about $S^n$ by passing to $\mathbb{R}^n$ or vice versa.
Example. The sphere $S^n$ is simply connected. One way to prove this is to show that a path in $S^n$ can be deformed so that it misses one point (this is the hard step). From here, removing the missed point gives a path in $\mathbb{R}^n$, which can be deformed into a constant path using linear functions.
In addition, a general principle in mathematics is that things which are unique are probably important. The one-point compactification is a construction of this type: it is the unique minimal compactification (of a locally compact Hausdorff space).
Solution 2:
Consider a topological space . Then may or may not be compact depending on the topology . If is not compact, then we might want to construct a compact topological space from while preserving as much of the topology on as possible. This process can be achieved by compactification.
Please refer to the below link for visual example
http://mathonline.wikidot.com/one-point-compactification-of-a-topological-space