reserve T   for TopSpace,
        A,B for Subset of T;
reserve NT,NTX,NTY for NTopSpace,
        A,B        for Subset of NT,
        O          for open Subset of NT,
        a          for Point of NT,
        XA         for Subset of NTX,
        YB         for Subset of NTY,
        x          for Point of NTX,
        y          for Point of NTY,
        f          for Function of NTX,NTY,
        fc         for continuous Function of NTX,NTY;
reserve NT for T_2 NTopSpace;
reserve T   for non empty TopSpace,
        A,B for Subset of T,
        F   for closed Subset of T,
        O   for open Subset of T;
reserve T   for non empty strict TopSpace,
        A,B for Subset of T,
        x   for Point of T;
reserve A for Subset of FMT_R^1,
        x for Point of FMT_R^1,
        y for Point of RealSpace,
        z for Point of TopSpaceMetr RealSpace,
        r for Real;

theorem Th55:
  A is open iff for x being Real st x in A
  ex r st r > 0 & ].x - r, x + r.[ c= A
  proof
    A is Subset of NTop2Top Top2NTop R^1 by FINTOPO7:def 16;
    then reconsider A9 = A as Subset of R^1 by FINTOPO7:24;
    hereby
      assume A is open;
      then A is open Subset of NTop2Top FMT_R^1 &
        NTop2Top FMT_R^1 = R^1 by FINTOPO7:24,Lm9;
      hence for x being Real st x in A
      ex r being Real st r > 0 & ].x - r, x + r.[ c= A by FRECHET:8;
    end;
    assume for x being Real st x in A
      ex r being Real st r > 0 & ].x - r, x + r.[ c= A;
    then A9 is open by FRECHET:8;
    hence A is open by Lm1;
  end;
