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

theorem
  for f being Function of TOP-REAL m,R^1 holds f is open iff
  for p being Point of TOP-REAL m, r being positive Real
  ex s being positive Real st ].f.p-s,f.p+s.[ c= f.:Ball(p,r)
  proof
    let f be Function of TOP-REAL m,R^1;
A1: m in NAT by ORDINAL1:def 12;
    hereby
      assume
A2:   f is open;
      let p be Point of TOP-REAL m, r be positive Real;
      p in Ball(p,r) by A1,TOPGEN_5:13;
      then
A3:   f.p in f.:Ball(p,r) by FUNCT_2:35;
      f.:Ball(p,r) is open by A2;
      then consider s being Real such that
A4:   s > 0 and
A5:   ].f.p-s,f.p+s.[ c= f.:Ball(p,r) by A3,FRECHET:8;
      reconsider s as positive Real by A4;
      take s;
      thus ].f.p-s,f.p+s.[ c= f.:Ball(p,r) by A5;
    end;
    assume
A6: for p being Point of TOP-REAL m, r being positive Real
    ex s being positive Real st ].f.p-s,f.p+s.[ c= f.:Ball(p,r);
    let A be Subset of TOP-REAL m such that
A7: A is open;
    for x being set holds x in f.:A iff ex Q being Subset of R^1
    st Q is open & Q c= f.:A & x in Q
    proof
      let x be set;
      hereby
        assume x in f.:A;
        then consider p being Point of TOP-REAL m such that
A8:     p in A and
A9:     x = f.p by FUNCT_2:65;
        reconsider u = p as Point of Euclid m by EUCLID:67;
        A = Int A by A7,TOPS_1:23;
        then consider e being Real such that
A10:     e > 0 and
A11:     Ball(u,e) c= A by A8,GOBOARD6:5;
A12:     Ball(u,e) = Ball(p,e) by TOPREAL9:13;
        consider s being positive Real such that
A13:     ].f.p-s,f.p+s.[ c= f.:Ball(p,e) by A6,A10;
        take Q = R^1(].f.p-s,f.p+s.[);
        thus Q is open;
        f.:Ball(p,e) c= f.:A by A11,A12,RELAT_1:123;
        hence Q c= f.:A by A13;
        thus x in Q by A9,TOPREAL6:15;
      end;
      thus thesis;
    end;
    hence f.:A is open by TOPS_1:25;
  end;
