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

theorem
  X is discrete iff X is almost_discrete & for A being Subset of X, x
  being Point of X st A = {x} holds A is closed
proof
  thus X is discrete implies X is almost_discrete & for A being Subset of X, x
  being Point of X st A = {x} holds A is closed by Th16;
  assume
A1: X is almost_discrete;
  assume
A2: for A being Subset of X, x being Point of X st A = {x} holds A is closed;
  for A being Subset of X, x being Point of X st A = {x} holds A is open
  by A2,A1,Th22;
  hence thesis by Th17;
end;
