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

theorem Th22:
  X is almost_discrete iff for A being Subset of X st A is closed
  holds A is open
proof
  thus X is almost_discrete implies for A being Subset of X st A is closed
  holds A is open
  proof
    assume
A1: X is almost_discrete;
    let A be Subset of X;
    assume A is closed;
    then A` is closed by A1,Th21;
    hence thesis by TOPS_1:4;
  end;
  assume
A2: for A being Subset of X st A is closed holds A is open;
  now
    let A be Subset of X;
    assume A is open;
    then A` is open by A2;
    hence A is closed by TOPS_1:3;
  end;
  hence thesis by Th21;
end;
