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

theorem
  for Y being TopSpace-like TopStruct holds bool the carrier of Y = {{},
  the carrier of Y} implies Y is discrete & Y is anti-discrete
proof
  let Y be TopSpace-like TopStruct;
  reconsider E = {} as Subset-Family of Y by XBOOLE_1:2;
  reconsider E as Subset-Family of Y;
A1: the carrier of Y in the topology of Y by PRE_TOPC:def 1;
  E c= the topology of Y;
  then
A2: {} in the topology of Y by PRE_TOPC:def 1,ZFMISC_1:2;
  assume bool the carrier of Y = {{}, the carrier of Y};
  hence thesis by A2,A1,Th11;
end;
