reserve i, k, m, n for Nat,
  r, s for Real,
  rn for Real,
  x, y , z, X for set,
  T, T1, T2 for non empty TopSpace,
  p, q for Point of T,
  A, B, C for Subset of T,
  A9 for non empty Subset of T,
  pq for Element of [:the carrier of T,the carrier of T:],
  pq9 for Point of [:T,T:],
  pmet,pmet1 for Function of [:the carrier of T,the carrier of T:],REAL,
  pmet9,pmet19 for RealMap of [:T,T:] ,
  f,f1 for RealMap of T,
  FS2 for Functional_Sequence of [:the carrier of T,the carrier of T:],REAL,
  seq for Real_Sequence;

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;
