A topological space has a one-point compactification if and only if it is locally compact.
To see a part of this, assume is compact, , and . Let be a compact neighborhood of (relative to ), not containing . Then is also compact relative to , which shows is locally compact.
The point is often called the point of infinity.
A one-point compactification opens up for simplifications in definitions and proofs.
The continuous functions on may be of importance. Their restriction to are loosely the continuous functions on with a limit at infinity.