reserve T for non empty TopSpace,
  A, B for Subset of T,
  F, G for Subset-Family of T;

theorem
  T is separable iff ex A being Subset of T st A is dense countable
proof
  hereby
    consider A being Subset of T such that
A1: A is dense and
A2: density T = card A by TOPGEN_1:def 12;
    assume T is separable;
    then density T c= omega by TOPGEN_1:def 13;
    then A is countable by A2;
    hence ex A being Subset of T st A is dense countable by A1;
  end;
  given A being Subset of T such that
A3: A is dense countable;
  density T c= card A & card A c= omega by A3,TOPGEN_1:def 12;
  then density T c= omega;
  hence thesis by TOPGEN_1:def 13;
end;
