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

theorem Th5:
  for f being Function of TopSpaceMetr(M),T holds f is open iff
  for p being Point of M, r being positive Real
  ex W being open Subset of T st f.p in W & W c= f.:Ball(p,r)
  proof
    let f be Function of TopSpaceMetr(M),T;
    hereby
      assume
A1:   f is open;
      let p be Point of M, r be positive Real;
A2:   p in Ball(p,r) by GOBOARD6:1;
      reconsider V = Ball(p,r) as Subset of TopSpaceMetr(M);
      V is open by TOPMETR:14;
      then consider W being open Subset of T such that
A3:   f.p in W & W c= f.:V by A1,A2,Th3;
      take W;
      thus f.p in W & W c= f.:Ball(p,r) by A3;
    end;
    assume
A4: for p being Point of M, r being positive Real
    ex W being open Subset of T st f.p in W & W c= f.:Ball(p,r);
    for p being Point of TopSpaceMetr(M),
        V being open Subset of TopSpaceMetr(M) st p in V
    ex W being open Subset of T st f.p in W & W c= f.:V
    proof
      let p be Point of TopSpaceMetr(M);
      let V be open Subset of TopSpaceMetr(M);
      reconsider q = p as Point of M;
      assume p in V;
      then consider r being Real such that
A5:   r > 0 and
A6:   Ball(q,r) c= V by TOPMETR:15;
      consider W being open Subset of T such that
A7:   f.p in W and
A8:   W c= f.:Ball(q,r) by A4,A5;
      take W;
      thus f.p in W by A7;
      f.:Ball(q,r) c= f.:V by A6,RELAT_1:123;
      hence thesis by A8;
    end;
    hence thesis by Th3;
end;
