reserve
  n, m for Nat,
  T for non empty TopSpace,
  M, M1, M2 for non empty MetrSpace;

theorem
  for P being Subset of TOP-REAL m holds P is open iff
  for p being Point of TOP-REAL m st p in P
  ex r being positive Real st Ball(p,r) c= P
  proof
    let P be Subset of TOP-REAL m;
A1: the TopStruct of TOP-REAL m = TopSpaceMetr Euclid m by EUCLID:def 8;
    then
    reconsider P1 = P as Subset of TopSpaceMetr(Euclid m);
    hereby
      assume
A2:   P is open;
      let p be Point of TOP-REAL m;
      assume
A3:   p in P;
      reconsider e = p as Point of Euclid m by EUCLID:67;
      P1 is open by A2,A1,TOPS_3:76;
      then consider r being Real such that
A4:   r > 0 and
A5:   Ball(e,r) c= P1 by A3,TOPMETR:15;
      reconsider r as positive Real by A4;
      take r;
      thus Ball(p,r) c= P by A5,TOPREAL9:13;
    end;
    assume
A6: for p being Point of TOP-REAL m st p in P
    ex r being positive Real st Ball(p,r) c= P;
    for p being Point of Euclid m st p in P1
    ex r being Real st r > 0 & Ball(p,r) c= P1
    proof
      let p be Point of Euclid m;
      assume
A7:   p in P1;
      reconsider e = p as Point of TOP-REAL m by EUCLID:67;
      consider r being positive Real such that
A8:   Ball(e,r) c= P1 by A6,A7;
      take r;
      thus thesis by A8,TOPREAL9:13;
    end;
    then P1 is open by TOPMETR:15;
    hence thesis by A1,TOPS_3:76;
  end;
