reserve i,n,m for Nat,
  x,y,X,Y for set,
  r,s for Real;

theorem Th27:
  for T be T_1 non empty TopSpace holds T is countably_compact iff
  for A be Subset of T st A is infinite holds Der A is non empty
proof
  let T be T_1 non empty TopSpace;
  thus T is countably_compact implies for A be Subset of T st A is infinite
  holds Der A is non empty
  proof
    assume T is countably_compact;
    then for F be Subset-Family of T st F is locally_finite & for A be Subset
    of T st A in F holds card A=1 holds F is finite by Th26;
    hence thesis by Lm4;
  end;
  assume for A be Subset of T st A is infinite holds Der A is non empty;
  then
  for A be Subset of T st A is infinite countable holds Der A is non empty;
  hence thesis by Lm5;
end;
