reserve x,y,X for set;

theorem Th33:
  for T being non empty TopSpace holds T is compact iff for N
  being net of T ex x being Point of T st x is_a_cluster_point_of N
proof
  let T be non empty TopSpace;
  thus T is compact implies for N being net of T ex x being Point of T st x
  is_a_cluster_point_of N by Lm1;
  assume
  for N being net of T ex x being Point of T st x is_a_cluster_point_of N;
  then for N being net of T st N in NetUniv T ex x being Point of T st x
  is_a_cluster_point_of N;
  hence thesis by Lm2;
end;
