Theorem
An infinite subset
of a discrete space
is not compact.
Proof
Letbe a discrete topological space and letbe an infinite subset of
Consider the set of singleton sets
It is an open cover ofbecause 
and
is open inbecause every singleton set is open in the discrete topology.
No proper subset of
is an open cover of A. Since A is infinite so isand the open cover contains no fininite subcover andis not compact.
Obviously ifis a finite subset of a discrete space it is compact.