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

theorem Th16:
  X is discrete iff for A being Subset of X holds A is closed
proof
  thus X is discrete implies for A being Subset of X holds A is closed
  proof
    assume
A1: X is discrete;
    let A be Subset of X;
    A` is open by A1,Th15;
    hence thesis by TOPS_1:3;
  end;
  assume
A2: for A being Subset of X holds A is closed;
  now
    let A be Subset of X;
    A` is closed by A2;
    hence A is open by TOPS_1:4;
  end;
  hence thesis by Th15;
end;
