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

theorem Th18:
  X is anti-discrete iff for A being Subset of X st A is open
  holds A = {} or A = the carrier of X
proof
A1: the carrier of X in the topology of X by PRE_TOPC:def 1;
  thus X is anti-discrete implies for A being Subset of X st A is open holds A
  = {} or A = the carrier of X
  proof
    assume
A2: X is anti-discrete;
    let A be Subset of X;
    assume A is open;
    then A in the topology of X by PRE_TOPC:def 2;
    then A in {{}, the carrier of X} by A2;
    hence thesis by TARSKI:def 2;
  end;
  assume
A3: for A being Subset of X st A is open holds A = {} or A = the carrier of X;
  now
    let P be object;
    assume
A4: P in the topology of X;
    then reconsider C = P as Subset of X;
    C is open by A4,PRE_TOPC:def 2;
    then C = {} or C = the carrier of X by A3;
    hence P in {{}, the carrier of X} by TARSKI:def 2;
  end;
  then
A5: the topology of X c= {{}, the carrier of X};
  {} in the topology of X by PRE_TOPC:1;
  then {{}, the carrier of X} c= the topology of X by A1,ZFMISC_1:32;
  then the topology of X = {{}, the carrier of X} by A5;
  hence thesis;
end;
