reserve X for non empty TopSpace;
reserve Y for non empty TopStruct;
reserve x for Point of Y;
reserve Y for non empty TopStruct;
reserve X for non empty TopSpace;
reserve x,y for Point of X;
reserve A, B for Subset of X;
reserve P, Q for Subset of X;
reserve Y for non empty TopStruct;

theorem Th66:
  for Y0 being SubSpace of Y, A being Subset of Y st A = the
  carrier of Y0 holds Y0 is anti-discrete implies A is anti-discrete
proof
  let Y0 be SubSpace of Y, A be Subset of Y;
  assume
A1: A = the carrier of Y0;
  assume Y0 is anti-discrete;
  then
A2: the topology of Y0 = {{}, the carrier of Y0} by TDLAT_3:def 2;
  now
    let G be Subset of Y;
    reconsider H = A /\ G as Subset of Y0 by A1,XBOOLE_1:17;
    assume
A3: G is open;
    G in the topology of Y & H = G /\ [#]Y0 by A1,A3,PRE_TOPC:def 2;
    then H in the topology of Y0 by PRE_TOPC:def 4;
    then A /\ G = {} or A /\ G = the carrier of Y0 by A2,TARSKI:def 2;
    hence A misses G or A c= G by A1,XBOOLE_1:17;
  end;
  hence thesis;
end;
