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

theorem Th21:
  X is almost_discrete iff for A being Subset of X st A is open
  holds A is closed
proof
  thus X is almost_discrete implies for A being Subset of X st A is open holds
  A is closed
  proof
    assume
A1: X is almost_discrete;
    now
      let A be Subset of X;
      assume A is open;
      then A in the topology of X by PRE_TOPC:def 2;
      then (the carrier of X) \ A in the topology of X by A1;
      then [#]X \ A is open by PRE_TOPC:def 2;
      hence A is closed by PRE_TOPC:def 3;
    end;
    hence thesis;
  end;
  assume
A2: for A being Subset of X st A is open holds A is closed;
  now
    let A be Subset of X;
    reconsider A9 = A as Subset of X;
    assume A in the topology of X;
    then A9 is open by PRE_TOPC:def 2;
    then A9 is closed by A2;
    then [#]X \ A9 is open by PRE_TOPC:def 3;
    hence (the carrier of X) \ A in the topology of X by PRE_TOPC:def 2;
  end;
  hence thesis;
end;
