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.
