A noncompact space is a topological space that is not a compact space. Equivalently, it has an open cover with no finite subcover (Hatcher, p. 27).
The real line is noncompact. For example, the open intervals 
, for positive
integers 
, cover the real line,
but no finite collection of these intervals does. Noncompactness
does not imply unboundedness. The open interval 
is also noncompact, as shown by
its open cover consisting of 
for integers 
(Hatcher, p. 27).
