Available  (http://katlas.math.toronto.edu/0506-Topology/images/0/02/05-11-15.pdf)
An example of a limit-point compact topological space that it is not compact: Let and define for each n. Obviously the collection of all is a basis for a topology on . Notice that with this topology is not compact: the same family of all is an open cover of with no finite subcover. But is limit-point compact: suppose that is an infinite subset of and . Define x = 2n − 1 if a is even, and x = 2n if a is odd. Then every open set that contains x also contains a, so x is an accumulation point of .