reserve X for TopSpace;
reserve C for Subset of X;
reserve A, B for Subset of X;

theorem
  X is almost_discrete iff for A being Subset of X st A is open holds Cl A = A
proof
  thus X is almost_discrete implies for A being Subset of X st A is open holds
  Cl A = A
  by Th21,PRE_TOPC:22;
  assume
A1: for A being Subset of X st A is open holds Cl A = A;
  now
    let A be Subset of X;
    assume A is open;
    then Cl A = A by A1;
    hence A is closed;
  end;
  hence thesis by Th21;
end;
