theorem
  for T holds (T is regular & T is T_1 & ex Bn being FamilySequence of T
  st Bn is Basis_sigma_discrete) iff T is metrizable
proof
  let T;
  now
    assume that
A1: T is regular & T is T_1 and
A2: ex Bn be FamilySequence of T st Bn is Basis_sigma_discrete;
    consider Bn be FamilySequence of T such that
A3: Bn is Basis_sigma_discrete by A2;
    Bn is sigma_discrete by A3,NAGATA_1:def 5;
    then
A4: Bn is sigma_locally_finite by NAGATA_1:12;
    Union Bn is Basis of T by A3,NAGATA_1:def 5;
    then Bn is Basis_sigma_locally_finite by A4,NAGATA_1:def 6;
    hence T is metrizable by A1,Th19;
  end;
  hence thesis by Th21,NAGATA_1:15;
end;
