reserve A, B for non empty set,
  A1, A2, A3 for non empty Subset of A;
reserve X for TopSpace;
reserve X for non empty TopSpace;
reserve X1, X2 for non empty SubSpace of X;
reserve X0, X1, X2, Y1, Y2 for non empty SubSpace of X;
reserve X, Y for non empty TopSpace;
reserve f for Function of X,Y;
reserve X,Y,Z for non empty TopSpace;
reserve f for Function of X,Y,
  g for Function of Y,Z;
reserve X, Y for non empty TopSpace,
  X0 for non empty SubSpace of X;
reserve f for Function of X,Y;
reserve f for Function of X,Y,
  X0 for non empty SubSpace of X;
reserve X, Y for non empty TopSpace,
  X0, X1 for non empty SubSpace of X;
reserve f for Function of X,Y,
  g for Function of X0,Y;
reserve X0, X1, X2 for non empty SubSpace of X;
reserve f for Function of X,Y,
  g for Function of X0,Y;
reserve X for non empty TopSpace,
  H, G for Subset of X;
reserve A for Subset of X;

theorem Th101:
  for A being Subset of X holds A is open iff modid(X,A) is
  continuous Function of X,X modified_with_respect_to A
proof
  let A be Subset of X;
  thus A is open implies modid(X,A) is continuous Function of X,X
  modified_with_respect_to A
  proof
    reconsider f = modid(X,A) as Function of X,X;
A1: f = id X;
    assume A is open;
    then the TopStruct of X = X modified_with_respect_to A by Th95;
    hence thesis by A1,Th51;
  end;
A2: [#](X modified_with_respect_to A) <> {};
  thus modid(X,A) is continuous Function of X,X modified_with_respect_to A
  implies A is open
  proof
    set B = (modid(X,A)).:A;
    assume
A3: modid(X,A) is continuous Function of X,X modified_with_respect_to A;
    B = A by FUNCT_1:92;
    then
A4: (modid(X,A))"B = A by FUNCT_2:94;
    B is open by Th94,FUNCT_1:92;
    hence thesis by A2,A3,A4,TOPS_2:43;
  end;
end;
