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

theorem Th11:
  for Y being TopStruct st {} in the topology of Y & the carrier
of Y in the topology of Y holds bool the carrier of Y = {{}, the carrier of Y}
  implies Y is discrete & Y is anti-discrete
proof
  let Y be TopStruct;
  assume that
A1: {} in the topology of Y and
A2: the carrier of Y in the topology of Y;
  assume
A3: bool the carrier of Y = {{}, the carrier of Y};
  {{}, the carrier of Y} c= the topology of Y by A1,A2,ZFMISC_1:32;
  then the topology of Y = bool the carrier of Y by A3;
  hence thesis by A3;
end;
